Package ‘VaRES’ · Computes Value at risk and expected shortfall, two most popular measures of ﬁnancial risk, for over one hundred parametric distributions, including all commonly

Package ‘VaRES’February 19, 2015

Type Package

Title Computes value at risk and expected shortfall for over 100parametric distributions

Version 1.0

Date 2013-8-25

Author Saralees Nadarajah, Stephen Chan and Emmanuel Afuecheta

Maintainer Saralees Nadarajah <[email protected]>

Depends R (>= 2.15.0)

Description Computes Value at risk and expected shortfall, two most popular measures of finan-cial risk, for over one hundred parametric distributions, including all commonly known distribu-tions. Also computed are the corresponding probability density function and cumulative distri-bution function.

License GPL (>= 2)

NeedsCompilation no

Repository CRAN

Date/Publication 2013-08-27 08:07:57

R topics documented:VaRES-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4aep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6ast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8asylaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9asypower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11beard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13betaburr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15betaburr7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16betadist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17betaexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19betafrechet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20betagompertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1

2 R topics documented:

betagumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23betagumbel2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24betalognorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25betalomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26betanorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28betapareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29betaweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30BS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32burr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33burr7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37clg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38compbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39dagum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41dweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42expexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44expext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45expgeo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46explog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47explogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50exppois . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51exppower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52expweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55FR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56frechet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59genbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60genbeta2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61geninvbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63genlogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64genlogis3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65genlogis4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67genpareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68genpowerweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69genunif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71gev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72gompertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75gumbel2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76halfcauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77halflogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78halfnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80halfT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81HBlaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82HL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

R topics documented: 3

Hlogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84invbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86invexpexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87invgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88kum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89kumburr7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91kumexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92kumgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93kumgumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95kumhalfnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96kumloglogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97kumnormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99kumpareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100kumweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103lfr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104LNbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106logbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107logcauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108loggamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109logisexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111logisrayleigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113loglaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114loglog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116loglogis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117lognorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119lomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Mlaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121moexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123moweibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124MRbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125nakagami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129paretostable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130PCTAlaplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132perks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133power1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134power2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136quad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137rgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138RS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140schabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142stacygamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144TL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4 aep

TL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148tsp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152xie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Index 156

VaRES-package Computes value at risk and expected shortfall for over 100 parametricdistributions

Description

Computes Value at risk and expected shortfall, two most popular measures of financial risk, for overone hundred parametric distributions, including all commonly known distributions. Also computedare the corresponding probability density function and cumulative distribution function.

Details

Package: VaRESType: PackageVersion: 1.0Date: 2013-8-25License: GPL(>=2)

Author(s)

Saralees Nadarajah, Stephen Chan and Emmanuel Afuecheta

Maintainer: Saralees Nadarajah <[email protected]>

References

S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall,submitted

aep Asymmetric exponential power distribution

aep 5

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric exponential powerdistribution due to Zhu and Zinde-Walsh (2009) given by

f(x) =

α

α∗K (q1) exp

[− 1

q1

∣∣∣ x2α∗

∣∣∣q1] , if x ≤ 0,

1− α1− α∗

K (q2) exp

[− 1

q2

∣∣∣∣ x

2− 2α∗

∣∣∣∣q2] , if x > 0

F (x) =

αQ

(1

q1

(| x |2α∗

)q1,

1

q1

), if x ≤ 0,

1− (1− α)Q

(1

q2

(| x |

2− 2α∗

)q2,

1

q2

), if x > 0

VaRp(X) =

−2α∗

[q1Q

−1

(p

α,

1

q1

)] 1q1

, if p ≤ α,

2 (1− α∗)[q2Q

−1

(1− p1− α

,1

q2

)] 1q2

, if p > α,

ESp(X) =

−2α∗

p

∫ p

0

[q1Q

−1

(v

α,

1

q1

)] 1q1

dv, if p ≤ α,

−2α∗

p

∫ α

0

[q1Q

−1

(v

α,

1

q1

)] 1q1

dv

+2 (1− α∗)

p

∫ p

α

[q2Q

−1

(1− v1− α

,1

q2

)] 1q2

dv, if p > α

for −∞ < x < ∞, 0 0, the first shape parame-ter, and q2 > 0, the second shape parameter, where α∗ = αK (q1) / {αK (q1) + (1− α)K (q2)},K(q) = 1

2q1/qΓ(1+1/q), Q(a, x) =

∫∞xta−1 exp (−t) dt/Γ(a) denotes the regularized complemen-

tary incomplete gamma function, Γ(a) =∫∞

0ta−1 exp (−t) dt denotes the gamma function, and

Q−1(a, x) denotes the inverse of Q(a, x).

Usage

daep(x, q1=1, q2=1, alpha=0.5, log=FALSE)paep(x, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)varaep(p, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)esaep(p, q1=1, q2=1, alpha=0.5)

Arguments

x scaler or vector of values at which the pdf or cdf needs to be computed

p scaler or vector of values at which the value at risk or expected shortfall needsto be computed

alpha the value of the scale parameter, must be in the unit interval, the default is 0.5

6 arcsine

q1 the value of the first shape parameter, must be positive, the default is 1

q2 the value of the second shape parameter, must be positive, the default is 1

log if TRUE then log(pdf) are returned

log.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)

lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of thesame length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)daep(x)paep(x)varaep(x)esaep(x)

arcsine Arcsine distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the arcsine distribution given by

f(x) =1

π√

(x− a)(b− x),

F (x) =2

πarcsin

(√x− ab− a

),

VaRp(X) = a+ (b− a) sin2(πp

2

),

ESp(X) = a+b− ap

∫ p

0

sin2(πv

2

)dv

for a ≤ x ≤ b, 0 < p < 1, −∞ < a <∞, the first location parameter, and −∞ < a < b <∞, thesecond location parameter.

arcsine 7

Usage

darcsine(x, a=0, b=1, log=FALSE)parcsine(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)vararcsine(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)esarcsine(p, a=0, b=1)

Arguments



a the value of the first location parameter, can take any real value, the default iszero

b the value of the second location parameter, can take any real value but must begreater than a, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)darcsine(x)parcsine(x)vararcsine(x)esarcsine(x)

8 ast

ast Generalized asymmetric Student’s t distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized asymmetric Student’st distribution due to Zhu and Galbraith (2010) given by

f(x) =

α

α∗K (ν1)

[1 +

1

ν1

( x

2α∗

)2]− ν1+1

2

, if x ≤ 0,

1− α1− α∗

K (ν2)

[1 +

1

ν2

(x

2 (1− α∗)

)2]− ν2+1

2

, if x > 0

F (x) = 2αFν1

(min(x, 0)

2α∗

)− 1 + α+ 2(1− α)Fν2

(max(x, 0)

2− 2α∗

),

VaRp(X) = 2α∗F−1ν1

(min(p, α)

2α

)+ 2 (1− α∗)F−1

ν2

(max(p, α) + 1− 2α

2− 2α

),

ESp(X) =2α∗

p

∫ p

0

F−1ν1

(min(v, α)

2α

)dv +

2 (1− α∗)p

∫ p

0

F−1ν2

(max(v, α) + 1− 2α

2− 2α

)dv

for −∞ < x < ∞, 0 0, the first de-gree of freedom parameter, and ν2 > 0, the second degree of freedom parameter, where α∗ =αK (ν1) / {αK (ν1) + (1− α)K (ν2)}, K(ν) = Γ ((ν + 1)/2) / [

√πνΓ(ν/2)], Fν(·) denotes the

cdf of a Student’s t random variable with ν degrees of freedom, and F−1ν (·) denotes the inverse of

Fν(·).

Usage

dast(x, nu1=1, nu2=1, alpha=0.5, log=FALSE)past(x, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)varast(p, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)esast(p, nu1=1, nu2=1, alpha=0.5)

Arguments

x scaler or vector of values at which the pdf or cdf needs to be computedp scaler or vector of values at which the value at risk or expected shortfall needs

to be computedalpha the value of the scale parameter, must be in the unit interval, the default is 0.5nu1 the value of the first degree of freedom parameter, must be positive, the default

is 1nu2 the value of the second degree of freedom parameter, must be positive, the de-

fault is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

asylaplace 9

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dast(x)past(x)varast(x)esast(x)

asylaplace Asymmetric Laplace distribution

10 asylaplace

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric Laplace distributiondue to Kotz et al. (2001) given by

f(x) =

κ√

2

τ (1 + κ2)exp

(−κ√

2

τ|x− θ|

), if x ≥ θ,

κ√

2

τ (1 + κ2)exp

(−√

2

κτ|x− θ|

), if x < θ,

F (x) =

1− 1

1 + κ2exp

(κ√

2(θ − x)

τ

), if x ≥ θ,

κ2

1 + κ2exp

(√2(x− θ)κτ

), if x < θ,

VaRp(X) =

θ − τ√

2κlog[(1− p)

(1 + κ2

)], if p ≥ κ2

1+κ2 ,

θ +κτ√

2log[p(1 + κ−2

)], if p < κ2

1+κ2 ,

ESp(X) =

θ

p+ θ − τ√

2κlog(1 + κ2

)+

√2τ(1 + 2κ2

)2κ (1 + κ2) p

log(1 + κ2

)−√

2τκ log κ

(1 + κ2) p− θκ2

(1 + κ2) p+

τ(1− κ4

)√

2κ (1 + κ2) p

−τ(1− p)√2κp

+τ(1− p)√

2κplog(1− p), if p ≥ κ2

1+κ2 ,

θ +κτ√

2log(1 + κ−2

)+κτ√

2(log p− 1), if p < κ2

1+κ2

for −∞ < x < ∞, 0 0, the first scaleparameter, and τ > 0, the second scale parameter.

Usage

dasylaplace(x, tau=1, kappa=1, theta=0, log=FALSE)pasylaplace(x, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)varasylaplace(p, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)esasylaplace(p, tau=1, kappa=1, theta=0)

Arguments



theta the value of the location parameter, can take any real value, the default is zero

kappa the value of the first scale parameter, must be positive, the default is 1

asypower 11

tau the value of the second scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dasylaplace(x)pasylaplace(x)varasylaplace(x)esasylaplace(x)

asypower Asymmetric power distribution

12 asypower

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric power distributiondue to Komunjer (2007) given by

f(x) =

δ1/λ

Γ(1 + 1/λ)exp

[− δ

aλ|x|λ

], if x ≤ 0,

δ1/λ

Γ(1 + 1/λ)exp

[− δ

(1− a)λ|x|λ

], if x > 0

F (x) =

a− aI

(δ

aλ

√λ|x|λ, 1/λ

), if x ≤ 0,

a− (1− a)I(

δ

(1− a)λ

√λ|x|λ, 1/λ

), if x > 0

VaRp(X) =

−[aλ

δ√λ

]1/λ [I−1

(1− p

a,

1

λ

)]1/λ

, if p ≤ a,

−[

(1− a)λ

δ√λ

]1/λ [I−1

(1− 1− p

1− a,

1

λ

)]1/λ

, if p > a,

ESp(X) =

−1

p

[aλ

δ√λ

]1/λ ∫ p

0

[I−1

(1− v

a,

1

λ

)]1/λ

dv, if p ≤ a,

−1

p

[aλ

δ√λ

]1/λ ∫ a

0

[I−1

(1− v

a,

1

λ

)]1/λ

dv

−1

p

[(1− a)λ

δ√λ

]1/λ ∫ p

a

[I−1

(1− 1− v

1− a,

1

λ

)]1/λ

dv, if p > a

for −∞ < x < ∞, 0 0, the second scaleparameter, and λ > 0, the shape parameter, where I(x, γ) = 1

Γ(γ)

∫ x√γ0

tγ−1 exp(−t)dt.

Usage

dasypower(x, a=0.5, lambda=1, delta=1, log=FALSE)pasypower(x, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)varasypower(p, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)esasypower(p, a=0.5, lambda=1, delta=1)

Arguments



a the value of the first scale parameter, must be in the unit interval, the default is0.5

delta the value of the second scale parameter, must be positive, the default is 1

lambda the value of the shape parameter, must be positive, the default is 1

beard 13




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dasypower(x)pasypower(x)varasypower(x)esasypower(x)

beard Beard distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Beard distribution due to Beard(1959) given by

f(x) =a exp(bx) [1 + aρ]

ρ−1/b

[1 + aρ exp(bx)]1+ρ−1/b

,

F (x) = 1− [1 + aρ]ρ−1/b

[1 + aρ exp(bx)]ρ−1/b

,

VaRp(X) =1

blog

[1 + aρ

aρ(1− p)ρ1/b− 1

aρ

],

ESp(X) =1

pb

∫ p

0

log

[− 1

aρ+

1 + aρ

aρ(1− v)ρ1/b

]dv

for x > 0, 0 0, the first scale parameter, b > 0, the second scale parameter, andρ > 0, the shape parameter.

14 beard

Usage

dbeard(x, a=1, b=1, rho=1, log=FALSE)pbeard(x, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)varbeard(p, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)esbeard(p, a=1, b=1, rho=1)

Arguments



a the value of the first scale parameter, must be positive, the default is 1

b the value of the second scale parameter, must be positive, the default is 1

rho the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbeard(x)pbeard(x)varbeard(x)esbeard(x)

betaburr 15

betaburr Beta Burr distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr distribution due toParana\’iba et al. (2011) given by

f(x) =babd

B(c, d)xbd+1

[1 + (x/a)

−b]−c−d

,

F (x) = I 1

1+(x/a)−b(c, d),

VaRp(X) = a[I−1p (c, d)

]1/b [1− I−1

p (c, d)]−1/b

,

ESp(X) =a

p

∫ p

0

[I−1v (c, d)

]1/b [1− I−1

v (c, d)]−1/b

dv

for x > 0, 0 0, the scale parameter, b > 0, the first shape parameter, c > 0,the second shape parameter, and d > 0, the third shape parameter, where Ix(a, b) =

∫ x0ta−1(1 −

t)b−1dt/B(a, b) denotes the incomplete beta function ratio,B(a, b) =∫ 1

0ta−1(1−t)b−1dt denotes

the beta function, and I−1x (a, b) denotes the inverse function of Ix(a, b).

Usage

dbetaburr(x, a=1, b=1, c=1, d=1, log=FALSE)pbetaburr(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varbetaburr(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esbetaburr(p, a=1, b=1, c=1, d=1)

Arguments



a the value of the scale parameter, must be positive, the default is 1

b the value of the first shape parameter, must be positive, the default is 1

c the value of the second shape parameter, must be positive, the default is 1

d the value of the third shape parameter, must be positive, the default is 1




Value


16 betaburr7

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetaburr(x)pbetaburr(x)varbetaburr(x)esbetaburr(x)

betaburr7 Beta Burr XII distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr XII distribution givenby

f(x) =kcxc−1

B(a, b)

[1− (1 + xc)

−k]a−1

(1 + xc)−bk−1

,

F (x) = I1−(1+xc)−k(a, b),

VaRp(X) ={[

1− I−1p (a, b)

]−1/k − 1}1/c

,

ESp(X) =1

p

∫ p

0

{[1− I−1

v (a, b)]−1/k − 1

}1/c

dv

for x > 0, 0 0, the first shape parameter, b > 0, the second shape parameter, c > 0,the third shape parameter, and k > 0, the fourth shape parameter.

Usage

dbetaburr7(x, a=1, b=1, c=1, k=1, log=FALSE)pbetaburr7(x, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE)varbetaburr7(p, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE)esbetaburr7(p, a=1, b=1, c=1, k=1)

Arguments



a the value of the first shape parameter, must be positive, the default is 1

betadist 17

b the value of the second shape parameter, must be positive, the default is 1

c the value of the third shape parameter, must be positive, the default is 1

k the value of the fourth shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetaburr7(x)pbetaburr7(x)varbetaburr7(x)esbetaburr7(x)

betadist Beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta distribution given by

f(x) =xa−1(1− x)b−1

B(a, b),

F (x) = Ix(a, b),VaRp(X) = I−1

p (a, b),

ESp(X) =1

p

∫ p

0

I−1v (a, b)dv

for 0 < x < 1, 0 0, the first parameter, and b > 0, the second shape parameter.

18 betadist

Usage

dbetadist(x, a=1, b=1, log=FALSE)pbetadist(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varbetadist(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esbetadist(p, a=1, b=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetadist(x)pbetadist(x)varbetadist(x)esbetadist(x)

betaexp 19

betaexp Beta exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta exponential distribution dueto Nadarajah and Kotz (2006) given by

f(x) =λ exp(−bλx)

B(a, b)[1− exp(−λx)]

a−1,

F (x) = I1−exp(−λx)(a, b),

VaRp(X) = − 1

λlog[1− I−1

p (a, b)],

ESp(X) = − 1

pλ

∫ p

0

log[1− I−1

v (a, b)]dv

for x > 0, 0 0, the first shape parameter, b > 0, the second shape parameter, andλ > 0, the scale parameter, where Ix(a, b) =

∫ x0ta−1(1− t)b−1dt/B(a, b) denotes the incomplete

beta function ratio, B(a, b) =∫ 1

0ta−1(1− t)b−1dt denotes the beta function, and I−1

x (a, b) denotesthe inverse function of Ix(a, b).

Usage

dbetaexp(x, lambda=1, a=1, b=1, log=FALSE)pbetaexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varbetaexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esbetaexp(p, lambda=1, a=1, b=1)

Arguments



lambda the value of the scale parameter, must be positive, the default is 1






Value


20 betafrechet

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetaexp(x)pbetaexp(x)varbetaexp(x)esbetaexp(x)

betafrechet Beta Frechet distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Fr\’echet distribution due toBarreto-Souza et al. (2011) given by

f(x) =ασα

xα+1B(a, b)exp

{−a(σx

)α}[1− exp

{−(σx

)α}]b−1

,

F (x) = Iexp{−(σx )α}(a, b),

VaRp(X) = σ[− log I−1

p (a, b)]−1/α

,

ESp(X) =σ

p

∫ p

0

[− log I−1

v (a, b)]−1/α

dv

for x > 0, 0 0, the first shape parameter, σ > 0, the scale parameter, b > 0, thesecond shape parameter, and α > 0, the third shape parameter.

Usage

dbetafrechet(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)pbetafrechet(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetafrechet(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetafrechet(p, a=1, b=1, alpha=1, sigma=1)

Arguments



sigma the value of the scale parameter, must be positive, the default is 1

betagompertz 21



alpha the value of the third shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetafrechet(x)pbetafrechet(x)varbetafrechet(x)esbetafrechet(x)

betagompertz Beta Gompertz distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Gompertz distribution dueto Cordeiro et al. (2012b) given by

f(x) =bη exp(bx)

B(c, d)exp (dη) exp [−dη exp(bx)] {1− exp [η − η exp(bx)]}c−1

,

F (x) = I1−exp[η−η exp(bx)](c, d),

VaRp(X) =1

blog

{1− 1

ηlog[1− I−1

p (c, d)]}

,

ESp(X) =1

pb

∫ p

0

log

{1− 1

ηlog[1− I−1

v (c, d)]}

dv

for x > 0, 0 0, the first scale parameter, η > 0, the second scale parameter, c > 0,the first shape parameter, and d > 0, the second shape parameter.

22 betagompertz

Usage

dbetagompertz(x, b=1, c=1, d=1, eta=1, log=FALSE)pbetagompertz(x, b=1, c=1, d=1, eta=1, log.p=FALSE, lower.tail=TRUE)varbetagompertz(p, b=1, c=1, d=1, eta=1, log.p=FALSE, lower.tail=TRUE)esbetagompertz(p, b=1, c=1, d=1, eta=1)

Arguments



b the value of the first scale parameter, must be positive, the default is 1

eta the value of the second scale parameter, must be positive, the default is 1

c the value of the first shape parameter, must be positive, the default is 1

d the value of the second shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetagompertz(x)pbetagompertz(x)varbetagompertz(x)esbetagompertz(x)

betagumbel 23

betagumbel Beta Gumbel distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Gumbel distribution due toNadarajah and Kotz (2004) given by

f(x) =1

σB(a, b)exp

(µ− xσ

)exp

[−a exp

µ− xσ

]{1− exp

[− exp

µ− xσ

]}b−1

,

F (x) = Iexp[− exp µ−xσ ](a, b),

VaRp(X) = µ− σ log[− log I−1

p (a, b)],

ESp(X) = µ− σ

p

∫ p

0

log[− log I−1

v (a, b)]dv

for −∞ < x <∞, 0 0, the scale parameter,a > 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dbetagumbel(x, a=1, b=1, mu=0, sigma=1, log=FALSE)pbetagumbel(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetagumbel(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetagumbel(p, a=1, b=1, mu=0, sigma=1)

Arguments



mu the value of the location parameter, can take any real value, the default is zero







Value


Author(s)

Saralees Nadarajah

24 betagumbel2

References


Examples

x=runif(10,min=0,max=1)dbetagumbel(x)pbetagumbel(x)varbetagumbel(x)esbetagumbel(x)

betagumbel2 Beta Gumbel 2 distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Gumbel II distribution givenby

f(x) =abx−a−1

B(c, d)exp

(−bdx−a

) [1− exp

(−bx−a

)]c−1,

F (x) = I1−exp(−bx−a)(c, d),

VaRp(X) = b1/a{− log

[1− I−1

p (c, d)]}−1/a

,

ESp(X) =b1/a

p

∫ p

0

{− log

[1− I−1

v (c, d)]}−1/a

dv

for x > 0, 0 0, the first shape parameter, b > 0, the scale parameter, c > 0, thesecond shape parameter, and d > 0, the third shape parameter.

Usage

dbetagumbel2(x, a=1, b=1, c=1, d=1, log=FALSE)pbetagumbel2(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varbetagumbel2(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esbetagumbel2(p, a=1, b=1, c=1, d=1)

Arguments


to be computedb the value of the scale parameter, must be positive, the default is 1a the value of the first shape parameter, must be positive, the default is 1c the value of the second shape parameter, must be positive, the default is 1d the value of the third shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

betalognorm 25

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetagumbel2(x)pbetagumbel2(x)varbetagumbel2(x)#esbetagumbel2(x)

betalognorm Beta lognormal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta lognormal distribution dueto Castellares et al. (2013) given by

f(x) =1

σxB(a, b)φ

(log x− µ

σ

)Φa−1

(log x− µ

σ

)Φb−1

(µ− log x

σ

),

F (x) = IΦ( log x−µσ )(a, b),

VaRp(X) = exp[µ+ σΦ−1

(I−1p (a, b)

)],

ESp(X) =exp(µ)

p

∫ p

0

exp[σΦ−1

(I−1v (a, b)

)]dv

for x > 0, 0 0, the scale parameter, a > 0,the first shape parameter, and b > 0, the second shape parameter, where φ(·) denotes the pdf of astandard normal random variable, and Φ(·) denotes the cdf of a standard normal random variable.

Usage

dbetalognorm(x, a=1, b=1, mu=0, sigma=1, log=FALSE)pbetalognorm(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetalognorm(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetalognorm(p, a=1, b=1, mu=0, sigma=1)

26 betalomax

Arguments










Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetalognorm(x)pbetalognorm(x)varbetalognorm(x)esbetalognorm(x)

betalomax Beta Lomax distribution

betalomax 27

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Lomax distribution due toLemonte and Cordeiro (2013) given by

f(x) =α

λB(a, b)

(1 +

x

λ

)−bα−1[1−

(1 +

x

λ

)−α]a−1

,

F (x) = I1−(1+ x

λ )−α(a, b),

VaRp(X) = λ[1− I−1

p (a, b)]−1/α − λ,

ESp(X) =λ

p

∫ p

0

[1− I−1

v (a, b)]−1/α

dv − λ

for x > 0, 0 0, the first shape parameter, b > 0, the second shape parameter, α > 0,the third shape parameter, and λ > 0, the scale parameter.

Usage

dbetalomax(x, a=1, b=1, alpha=1, lambda=1, log=FALSE)pbetalomax(x, a=1, b=1, alpha=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varbetalomax(p, a=1, b=1, alpha=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esbetalomax(p, a=1, b=1, alpha=1, lambda=1)

Arguments






alpha the value of the third scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


28 betanorm

Examples

x=runif(10,min=0,max=1)dbetalomax(x)pbetalomax(x)varbetalomax(x)esbetalomax(x)

betanorm Beta normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta normal distribution due toEugene et al. (2002) given by

f(x) =1

σB(a, b)φ

(x− µσ

)Φa−1

(x− µσ

)Φb−1

(µ− xσ

),

F (x) = IΦ( x−µσ )(a, b),

VaRp(X) = µ+ σΦ−1(I−1p (a, b)

),

ESp(X) = µ+σ

p

∫ p

0

Φ−1(I−1v (a, b)

)dv


Usage

dbetanorm(x, mu=0, sigma=1, a=1, b=1, log=FALSE)pbetanorm(x, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varbetanorm(p, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esbetanorm(p, mu=0, sigma=1, a=1, b=1)

Arguments










betapareto 29

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetanorm(x)pbetanorm(x)varbetanorm(x)esbetanorm(x)

betapareto Beta Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Pareto distribution due toAkinsete et al. (2008) given by

f(x) =aKadx−ad−1

B(c, d)

[1−

(K

x

)a]c−1

,

F (x) = I1−(Kx )a(c, d),

VaRp(X) = K[1− I−1

p (c, d)]−1/a

,

ESp(X) =K

p

∫ p

0

[1− I−1

v (c, d)]−1/a

dv

for x ≥ K, 0 0, the scale parameter, a > 0, the first shape parameter, c > 0, thesecond shape parameter, and d > 0, the third shape parameter.

Usage

dbetapareto(x, K=1, a=1, c=1, d=1, log=FALSE)pbetapareto(x, K=1, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varbetapareto(p, K=1, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esbetapareto(p, K=1, a=1, c=1, d=1)

30 betaweibull

Arguments



K the value of the scale parameter, must be positive, the default is 1







Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dbetapareto(x)pbetapareto(x)varbetapareto(x)esbetapareto(x)

betaweibull Beta Weibull distribution

betaweibull 31

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Weibull distribution due toCordeiro et al. (2012b) given by

f(x) =αxα−1

σαB(a, b)exp

{−b(xσ

)α}[1− exp

{−(xσ

)α}]a−1

,

F (x) = I1−exp{−( xσ )α}(a, b),

VaRp(X) = σ{− log

[1− I−1

p (a, b)]}1/α

,

ESp(X) =σ

p

∫ p

0

{− log

[1− I−1

v (a, b)]}1/α

dv

for x > 0, 0 0, the first shape parameter, b > 0, the second shape parameter, α > 0,the third shape parameter, and σ > 0, the scale parameter.

Usage

dbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)pbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varbetaweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esbetaweibull(p, a=1, b=1, alpha=1, sigma=1)

Arguments










Value


Author(s)

Saralees Nadarajah

References


32 BS

Examples

x=runif(10,min=0,max=1)dbetaweibull(x)pbetaweibull(x)varbetaweibull(x)esbetaweibull(x)

BS Birnbaum-Saunders distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Birnbaum-Saunders distributiondue to Birnbaum and Saunders (1969a, 1969b) given by

f(x) =x1/2 + x−1/2

2γxφ

(x1/2 − x−1/2

γ

),

F (x) = Φ

(x1/2 − x−1/2

γ

),

VaRp(X) =1

4

{γΦ−1(p) +

√4 + γ2 [Φ−1(p)]

2

}2

,

ESp(X) =1

4p

∫ p

0

{γΦ−1(v) +

√4 + γ2 [Φ−1(v)]

2

}2

dv

for x > 0, 0 0, the scale parameter.

Usage

dBS(x, gamma=1, log=FALSE)pBS(x, gamma=1, log.p=FALSE, lower.tail=TRUE)varBS(p, gamma=1, log.p=FALSE, lower.tail=TRUE)esBS(p, gamma=1)

Arguments



gamma the value of the scale parameter, must be positive, the default is 1




Value


burr 33

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dBS(x)pBS(x)varBS(x)esBS(x)

burr Burr distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Burr distribution due to Burr(1942) given by

f(x) =bab

xb+1

[1 + (x/a)

−b]−2

,

F (x) =1

1 + (x/a)−b ,

VaRp(X) = ap1/b(1− p)−1/b,

ESp(X) =a

pBp (1/b+ 1, 1− 1/b)

for x > 0, 0 0, the scale parameter, and b > 0, the shape parameter, whereBx(a, b) =

∫ x0ta−1(1− t)b−1dt denotes the incomplete beta function.

Usage

dburr(x, a=1, b=1, log=FALSE)pburr(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varburr(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esburr(p, a=1, b=1)

Arguments




b the value of the shape parameter, must be positive, the default is 1

34 burr7




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dburr(x)pburr(x)varburr(x)esburr(x)

burr7 Burr XII distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Burr XII distribution due to Burr(1942) given by

f(x) =kcxc−1

(1 + xc)k+1

,

F (x) = 1− (1 + xc)−k,

VaRp(X) =[(1− p)−1/k − 1

]1/c,

ESp(X) =1

p

∫ p

0

[(1− v)−1/k − 1

]1/cdv

for x > 0, 0 0, the first shape parameter, and k > 0, the second shape parameter.

Usage

dburr7(x, k=1, c=1, log=FALSE)pburr7(x, k=1, c=1, log.p=FALSE, lower.tail=TRUE)varburr7(p, k=1, c=1, log.p=FALSE, lower.tail=TRUE)esburr7(p, k=1, c=1)

Cauchy 35

Arguments



k the value of the first shape parameter, must be positive, the default is 1





Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dburr7(x)pburr7(x)varburr7(x)esburr7(x)

Cauchy Cauchy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Cauchy distribution given by

f(x) =1

π

σ

(x− µ)2 + σ2,

F (x) =1

2+

1

πarctan

(x− µσ

),

VaRp(X) = µ+ σ tan

(π

(p− 1

2

)),

ESp(X) = µ+σ

p

∫ p

0

tan

(π

(v − 1

2

))dv

36 Cauchy

for −∞ < x < ∞, 0 0, the scaleparameter.

Usage

dCauchy(x, mu=0, sigma=1, log=FALSE)pCauchy(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varCauchy(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esCauchy(p, mu=0, sigma=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dCauchy(x)pCauchy(x)varCauchy(x)esCauchy(x)

chen 37

chen Chen distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Chen distribution due to Chen(2000) given by

f(x) = λbxb−1 exp(xb)

exp[λ− λ exp

(xb)],

F (x) = 1− exp[λ− λ exp

(xb)],

VaRp(X) =

{log

[1− log(1− p)

λ

]}1/b

,

ESp(X) =1

p

∫ p

0

{log

[1− log(1− v)

λ

]}1/b

dv

for x > 0, 0 0, the shape parameter, and λ > 0, the scale parameter.

Usage

dchen(x, b=1, lambda=1, log=FALSE)pchen(x, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varchen(p, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)eschen(p, b=1, lambda=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


38 clg

Examples

x=runif(10,min=0,max=1)dchen(x)pchen(x)varchen(x)eschen(x)

clg Compound Laplace gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the compound Laplace gamma dis-tribution given by

f(x) =ab

2{1 + b |x− θ|}−(a+1)

,

F (x) =

1

2{1 + b |x− θ|}−a , if x ≤ θ,

1− 1

2{1 + b |x− θ|}−a , if x > θ,

VaRp(X) =

θ − 1

b− (2p)−1/a

b, if p ≤ 1/2,

θ − 1

b+

(2(1− p))−1/a

b, if p > 1/2,

ESp(X) =

θ − 1

b− (2p)−1/a

b(1− 1/a), if p ≤ 1/2,

θ − 1

b− [2(1− p)]1−1/a

2pb(1− 1/a), if p > 1/2

for −∞ < x <∞, 0 0, the scale parameter,and a > 0, the shape parameter.

Usage

dclg(x, a=1, b=1, theta=0, log=FALSE)pclg(x, a=1, b=1, theta=0, log.p=FALSE, lower.tail=TRUE)varclg(p, a=1, b=1, theta=0, log.p=FALSE, lower.tail=TRUE)esclg(p, a=1, b=1, theta=0)

Arguments



compbeta 39


b the value of the scale parameter, must be positive, the default is 1

a the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dclg(x)pclg(x)varclg(x)esclg(x)

compbeta Complementary beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the complementary beta distributiondue to Jones (2002) given by

f(x) = B(a, b){I−1x (a, b)

}1−a {1− I−1

x (a, b)}1−b

,F (x) = I−1

x (a, b),VaRp(X) = Ip(a, b),

ESp(X) =1

p

∫ p

0

Iv(a, b)dv

for 0 < x < 1, 0 0, the first shape parameter, and b > 0, the second shape parameter.

40 compbeta

Usage

dcompbeta(x, a=1, b=1, log=FALSE)pcompbeta(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varcompbeta(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)escompbeta(p, a=1, b=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dcompbeta(x)pcompbeta(x)varcompbeta(x)escompbeta(x)

dagum 41

dagum Dagum distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Dagum distribution due to Dagum(1975, 1977, 1980) given by

f(x) =acbaxac−1

[xa + ba]c+1 ,

F (x) =

[1 +

(b

x

)a]−c,

VaRp(X) = b(

1− p−1/c)−1/a

,

ESp(X) =b

p

∫ p

0

(1− v−1/c

)−1/a

dv

for x > 0, 0 0, the first shape parameter, b > 0, the scale parameter, and c > 0, thesecond shape parameter.

Usage

ddagum(x, a=1, b=1, c=1, log=FALSE)pdagum(x, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)vardagum(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esdagum(p, a=1, b=1, c=1)

Arguments









Value


42 dweibull

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)ddagum(x)pdagum(x)vardagum(x)esdagum(x)

dweibull Double Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the double Weibull distribution dueto Balakrishnan and Kocherlakota (1985) given by

f(x) =c

2σ

∣∣∣∣x− µσ∣∣∣∣c−1

exp

{−∣∣∣∣x− µσ

∣∣∣∣c} ,F (x) =

1

2exp

{−(µ− xσ

)c}, if x ≤ µ,

1− 1

2exp

{−(x− µσ

)c}, if x > µ,

VaRp(X) =

µ− σ [− log (2p)]1/c

, if p ≤ 1/2,

µ+ σ [− log (2(1− p))]1/c , if p > 1/2,

ESp(X) =

µ− σ

p

∫ p

0

[− log 2− log v]1/c

dv, if p ≤ 1/2,

µ− σ

p

∫ 1/2

0

[− log 2− log v]1/c

dv

+σ

p

∫ p

1/2

[− log 2− log(1− v)]1/c

dv, if p > 1/2

for −∞ < x <∞, 0 0, the scale parameter,and c > 0, the shape parameter.

dweibull 43

Usage

ddweibull(x, c=1, mu=0, sigma=1, log=FALSE)pdweibull(x, c=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vardweibull(p, c=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esdweibull(p, c=1, mu=0, sigma=1)

Arguments





c the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)ddweibull(x)pdweibull(x)vardweibull(x)esdweibull(x)

44 expexp

expexp Exponentiated exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated exponential distri-bution due to Gupta and Kundu (1999, 2001) given by

f(x) = aλ exp(−λx) [1− exp(−λx)]a−1

,F (x) = [1− exp(−λx)]

a,

VaRp(X) = − 1

λlog(

1− p1/a),

ESp(X) = − 1

pλ

∫ p

0

log(

1− v1/a)dv

for x > 0, 0 0, the shape parameter and λ > 0, the scale parameter.

Usage

dexpexp(x, lambda=1, a=1, log=FALSE)pexpexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varexpexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esexpexp(p, lambda=1, a=1)

Arguments


to be computedlambda the value of the scale parameter, must be positive, the default is 1a the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


expext 45

Examples

x=runif(10,min=0,max=1)dexpexp(x)pexpexp(x)varexpexp(x)esexpexp(x)

expext Exponential extension distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential extension distributiondue to Nadarajah and Haghighi (2011) given by

f(x) = aλ(1 + λx)a−1 exp [1− (1 + λx)a] ,F (x) = 1− exp [1− (1 + λx)a] ,

VaRp(X) =[1− log(1− p)]1/a − 1

λ,

ESp(X) = − 1

λ+

1

λp

∫ p

0

[1− log(1− v)]1/a

dv


Usage

dexpext(x, lambda=1, a=1, log=FALSE)pexpext(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varexpext(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esexpext(p, lambda=1, a=1)

Arguments








Value


46 expgeo

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dexpext(x)pexpext(x)varexpext(x)esexpext(x)

expgeo Exponential geometric distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential geometric distributiondue to Adamidis and Loukas (1998) given by

f(x) =λθ exp(−λx)

[1− (1− θ) exp(−λx)]2 ,

F (x) =θ exp(−λx)

1− (1− θ) exp(−λx),

VaRp(X) = − 1

λlog

p

θ + (1− θ)p,

ESp(X) = − log p

λ− θ log θ

λp(1− θ)+θ + (1− θ)pλp(1− θ)

log [θ + (1− θ)p]

for x > 0, 0 0, the second scale parameter.

Usage

dexpgeo(x, theta=0.5, lambda=1, log=FALSE)pexpgeo(x, theta=0.5, lambda=1, log.p=FALSE, lower.tail=TRUE)varexpgeo(p, theta=0.5, lambda=1, log.p=FALSE, lower.tail=TRUE)esexpgeo(p, theta=0.5, lambda=1)

Arguments



explog 47

theta the value of the first scale parameter, must be in the unit interval, the default is0.5

lambda the value of the second scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dexpgeo(x)pexpgeo(x)varexpgeo(x)esexpgeo(x)

explog Exponential logarithmic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential logarithmic distribu-tion due to Tahmasbi and Rezaei (2008) given by

f(x) = − b(1− a) exp(−bx)

log a [1− (1− a) exp(−bx)],

F (x) = 1− log [1− (1− a) exp(−bx)]

log a,

VaRp(X) = −1

blog

[1− a1−p

1− a

],

ESp(X) = − 1

bp

∫ p

0

log

[1− a1−v

1− a

]dv

for x > 0, 0 0, the second scale parameter.

48 explog

Usage

dexplog(x, a=0.5, b=1, log=FALSE)pexplog(x, a=0.5, b=1, log.p=FALSE, lower.tail=TRUE)varexplog(p, a=0.5, b=1, log.p=FALSE, lower.tail=TRUE)esexplog(p, a=0.5, b=1)

Arguments



a the value of the first scale parameter, must be in the unit interval, the default is0.5





Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dexplog(x)pexplog(x)varexplog(x)esexplog(x)

explogis 49

explogis Exponentiated logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated logistic distributiongiven by

f(x) = (a/b) exp(−x/b) [1 + exp(−x/b)]−a−1,

F (x) = [1 + exp(−x/b)]−a ,VaRp(X) = −b log

[p−1/a − 1

],

ESp(X) = − bp

∫ p

0

log[v−1/a − 1

]dv

for −∞ < x <∞, 0 0, the shape parameter, and b > 0, the scale parameter.

Usage

dexplogis(x, a=1, b=1, log=FALSE)pexplogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varexplogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esexplogis(p, a=1, b=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


50 exponential

Examples

x=runif(10,min=0,max=1)dexplogis(x)pexplogis(x)varexplogis(x)esexplogis(x)

exponential Exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential distribution given by

f(x) = λ exp(−λx),F (x) = 1− exp(−λx),

VaRp(X) = − 1

λlog(1− p),

ESp(X) = − 1

pλ{log(1− p)p− p− log(1− p)}

for x > 0, 0 0, the scale parameter.

Usage

dexponential(x, lambda=1, log=FALSE)pexponential(x, lambda=1, log.p=FALSE, lower.tail=TRUE)varexponential(p, lambda=1, log.p=FALSE, lower.tail=TRUE)esexponential(p, lambda=1)

Arguments







Value


Author(s)

Saralees Nadarajah

exppois 51

References


Examples

x=runif(10,min=0,max=1)dexponential(x)pexponential(x)varexponential(x)esexponential(x)

exppois Exponential Poisson distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential Poisson distributiondue to Kus (2007) given by

f(x) =bλ exp [−bx− λ+ λ exp(−bx)]

1− exp(−λ),

F (x) =1− exp [−λ+ λ exp(−bx)]

1− exp(−λ),

VaRp(X) = −1

blog

{1

λlog [1− p+ p exp(−λ)] + 1

},

ESp(X) = − 1

bp

∫ p

0

log

{1

λlog [1− v + v exp(−λ)] + 1

}dv

for x > 0, 0 0, the first scale parameter, and λ > 0, the second scale parameter.

Usage

dexppois(x, b=1, lambda=1, log=FALSE)pexppois(x, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varexppois(p, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esexppois(p, b=1, lambda=1)

Arguments


to be computedb the value of the first scale parameter, must be positive, the default is 1lambda the value of the second scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

52 exppower

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dexppois(x)pexppois(x)varexppois(x)esexppois(x)

exppower Exponential power distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential power distributiondue to Subbotin (1923) given by

f(x) =1

2a1/aσΓ (1 + 1/a)exp

{−| x− µ |

a

aσa

},

F (x) =

1

2Q

(1

a,

(µ− x)a

aσa

), if x ≤ µ,

1− 1

2Q

(1

a,

(x− µ)a

aσa

), if x > µ,

VaRp(X) =

µ− a1/aσ

[Q−1

(1

a, 2p

)]1/a

, if p ≤ 1/2,

µ+ a1/aσ[Q−1

(1a , 2(1− p)

)]1/a, if p > 1/2,

ESp(X) =

µ− a1/aσ

p

∫ p

0

[Q−1

(1

a, 2v

)]1/a

dv, if p ≤ 1/2,

µ− a1/aσ

p

∫ 1/2

0

[Q−1

(1

a, 2v

)]1/a

dv

+a1/aσ

p

∫ p

1/2

[Q−1

(1

a, 2(1− v)

)]1/a

dv, if p > 1/2

exppower 53

for −∞ < x <∞, 0 0, the scale parameter,and a > 0, the shape parameter.

Usage

dexppower(x, mu=0, sigma=1, a=1, log=FALSE)pexppower(x, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)varexppower(p, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)esexppower(p, mu=0, sigma=1, a=1)

Arguments









Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dexppower(x)pexppower(x)varexppower(x)esexppower(x)

54 expweibull

expweibull Exponentiated Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated Weibull distributiondue to Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) given by

f(x) = aασ−αxα−1 exp [−(x/σ)α] {1− exp [−(x/σ)α]}a−1,

F (x) = {1− exp [−(x/σ)α]}a ,

VaRp(X) = σ[− log

(1− p1/a

)]1/α,

ESp(X) =σ

p

∫ p

0

[− log

(1− v1/a

)]1/αdv

for x > 0, 0 0, the first shape parameter, α > 0, the second shape parameter, andσ > 0, the scale parameter.

Usage

dexpweibull(x, a=1, alpha=1, sigma=1, log=FALSE)pexpweibull(x, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varexpweibull(p, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esexpweibull(p, a=1, alpha=1, sigma=1)

Arguments





alpha the value of the second shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

F 55

References


Examples

x=runif(10,min=0,max=1)dexpweibull(x)pexpweibull(x)varexpweibull(x)esexpweibull(x)

F F distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the F distribution given by

f(x) =1

B(d12 ,

d22

) (d1

d2

) d12

xd12 −1

(1 +

d1

d2x

)− d1+d22

,

F (x) = I d1x

d1x+d2

(d1

2,d2

2

),

VaRp(X) =d2

d1

I−1p

(d12 ,

d22

)1− I−1

p

(d12 ,

d22

) ,ESp(X) =

d2

d1p

∫ p

0

I−1v

(d12 ,

d22

)1− I−1

v

(d12 ,

d22

)dvfor x ≥ K, 0 0, the first degree of freedom parameter, and d2 > 0, the seconddegree of freedom parameter.

Usage

dF(x, d1=1, d2=1, log=FALSE)pF(x, d1=1, d2=1, log.p=FALSE, lower.tail=TRUE)varF(p, d1=1, d2=1, log.p=FALSE, lower.tail=TRUE)esF(p, d1=1, d2=1)

Arguments



d1 the value of the first degree of freedom parameter, must be positive, the defaultis 1

56 FR

d2 the value of the second degree of freedom parameter, must be positive, the de-fault is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dF(x)pF(x)varF(x)esF(x)

FR Freimer distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Freimer distribution due toFreimer et al. (1988) given by

VaRp(X) =1

a

[pb − 1

b− (1− p)c − 1

c

],

ESp(X) =1

a

(1

c− 1

b

)+

pb

ab(b+ 1)+

(1− p)c+1 − 1

pac(c+ 1)

for 0 0, the scale parameter, b > 0, the first shape parameter, and c > 0, the secondshape parameter.

Usage

varFR(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esFR(p, a=1, b=1, c=1)

frechet 57

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)varFR(x)esFR(x)

frechet Frechet distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Fr\’echet distribution due toFr\’echet (1927) given by

f(x) =ασα

xα+1exp

{−(σx

)α},

F (x) = exp{−(σx

)α},

VaRp(X) = σ [− log p]−1/α

,

ESp(X) =σ

pΓ (1− 1/α,− log p)

for x > 0, 0 0, the shape parameter, and σ > 0, the scale parameter, whereΓ(a, x) =

∫∞xta−1 exp (−t) dt denotes the complementary incomplete gamma function.

58 frechet

Usage

dfrechet(x, alpha=1, sigma=1, log=FALSE)pfrechet(x, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varfrechet(p, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esfrechet(p, alpha=1, sigma=1)

Arguments




alpha the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dfrechet(x)pfrechet(x)varfrechet(x)esfrechet(x)

Gamma 59

Gamma Gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the gamma distribution given by

f(x) =baxa−1 exp(−bx)

Γ(a),

F (x) =γ(a, bx)

Γ(a),

VaRp(X) =1

bQ−1(a, 1− p),

ESp(X) =1

bp

∫ p

0

Q−1(a, 1− v)dv

for x > 0, 0 0, the scale parameter, and a > 0, the shape parameter, where γ(a, x) =∫ x0ta−1 exp (−t) dt denotes the incomplete gamma function,Q(a, x) =

∫∞xta−1 exp (−t) dt/Γ(a)

denotes the regularized complementary incomplete gamma function, Γ(a) =∫∞

0ta−1 exp (−t) dt

denotes the gamma function, and Q−1(a, x) denotes the inverse of Q(a, x).

Usage

dGamma(x, a=1, b=1, log=FALSE)pGamma(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varGamma(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esGamma(p, a=1, b=1)

Arguments








Value


Author(s)

Saralees Nadarajah

60 genbeta

References


Examples

x=runif(10,min=0,max=1)dGamma(x)pGamma(x)varGamma(x)esGamma(x)

genbeta Generalized beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized beta distributiongiven by

f(x) =(x− c)a−1(d− x)b−1

B(a, b)(d− c)a+b−1,

F (x) = I x−cd−c

(a, b),

VaRp(X) = c+ (d− c)I−1p (a, b),

ESp(X) = c+d− cp

∫ p

0

I−1v (a, b)dv

for c ≤ x ≤ d, 0 0, the first shape parameter, b > 0, the second shape parameter,−∞ < c <∞, the first location parameter, and −∞ < c < d <∞, the second location parameter.

Usage

dgenbeta(x, a=1, b=1, c=0, d=1, log=FALSE)pgenbeta(x, a=1, b=1, c=0, d=1, log.p=FALSE, lower.tail=TRUE)vargenbeta(p, a=1, b=1, c=0, d=1, log.p=FALSE, lower.tail=TRUE)esgenbeta(p, a=1, b=1, c=0, d=1)

Arguments



c the value of the first location parameter, can take any real value, the default iszero

d the value of the second location parameter, can take any real value but must begreater than c, the default is 1


genbeta2 61





Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgenbeta(x)pgenbeta(x)vargenbeta(x)esgenbeta(x)

genbeta2 Generalized beta II distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized beta II distributiongiven by

f(x) =cxac−1 (1− xc)b−1

B(a, b),

F (x) = Ixc(a, b),

VaRp(X) =[I−1p (a, b)

]1/c,

ESp(X) =1

p

∫ p

0

[I−1v (a, b)

]1/cdv

for 0 < x < 1, 0 0, the first shape parameter, b > 0, the second shape parameter, andc > 0, the third shape parameter.

62 genbeta2

Usage

dgenbeta2(x, a=1, b=1, c=1, log=FALSE)pgenbeta2(x, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)vargenbeta2(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esgenbeta2(p, a=1, b=1, c=1)

Arguments









Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgenbeta2(x)pgenbeta2(x)vargenbeta2(x)esgenbeta2(x)

geninvbeta 63

geninvbeta Generalized inverse beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized inverse beta distribu-tion given by

f(x) =axac−1

B(c, d) (1 + xa)c+d

,

F (x) = I xa

1+xa(c, d),

VaRp(X) =

[I−1p (c, d)

1− I−1p (c, d)

]1/a

,

ESp(X) =1

p

∫ p

0

[I−1v (c, d)

1− I−1v (c, d)

]1/a

dv

for x > 0, 0 0, the first shape parameter, c > 0, the second shape parameter, andd > 0, the third shape parameter.

Usage

dgeninvbeta(x, a=1, c=1, d=1, log=FALSE)pgeninvbeta(x, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)vargeninvbeta(p, a=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esgeninvbeta(p, a=1, c=1, d=1)

Arguments





d the value of the second shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

64 genlogis

References


Examples

x=runif(10,min=0,max=1)dgeninvbeta(x)pgeninvbeta(x)vargeninvbeta(x)esgeninvbeta(x)

genlogis Generalized logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic distributiongiven by

f(x) =a exp

(−x−µσ

)σ{

1 + exp(−x−µσ

)}1+a ,

F (x) =1{

1 + exp(−x−µσ

)}a ,VaRp(X) = µ− σ log

(p−1/a − 1

),

ESp(X) = µ− σ

p

∫ p

0

log(v−1/a − 1

)dv

for −∞ < x <∞, 0 0, the scale parameter,and a > 0, the shape parameter.

Usage

dgenlogis(x, a=1, mu=0, sigma=1, log=FALSE)pgenlogis(x, a=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargenlogis(p, a=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgenlogis(p, a=1, mu=0, sigma=1)

Arguments






genlogis3 65




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgenlogis(x)pgenlogis(x)vargenlogis(x)esgenlogis(x)

genlogis3 Generalized logistic III distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic III distributiongiven by

f(x) =1

σB(α, α)exp

(αx− µσ

){1 + exp

(x− µσ

)}−2α

,

F (x) = I 1

1+exp(− x−µσ )

(α, α),

VaRp(X) = µ− σ log1− I−1

p (α, α)

I−1p (α, α)

,

ESp(X) = µ− σ

p

∫ p

0

log1− I−1

v (α, α)

I−1v (α, α)

dv

for −∞ < x <∞, 0 0, the scale parameter,and α > 0, the shape parameter.

66 genlogis3

Usage

dgenlogis3(x, alpha=1, mu=0, sigma=1, log=FALSE)pgenlogis3(x, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargenlogis3(p, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgenlogis3(p, alpha=1, mu=0, sigma=1)

Arguments









Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgenlogis3(x)pgenlogis3(x)vargenlogis3(x)esgenlogis3(x)

genlogis4 67

genlogis4 Generalized logistic IV distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized logistic IV distributiongiven by

f(x) =1

σB(α, a)exp

(−αx− µ

σ

){1 + exp

(−x− µ

σ

)}−α−a,

F (x) = I 1

1+exp(− x−µσ )

(α, a),

VaRp(X) = µ− σ log1− I−1

p (α, a)

I−1p (α, a)

,

ESp(X) = µ− σ

p

∫ p

0

log1− I−1

v (α, a)

I−1v (α, a)

dv

for −∞ < x <∞, 0 0, the scale parameter,α > 0, the first shape parameter, and a > 0, the second shape parameter.

Usage

dgenlogis4(x, a=1, alpha=1, mu=0, sigma=1, log=FALSE)pgenlogis4(x, a=1, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargenlogis4(p, a=1, alpha=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgenlogis4(p, a=1, alpha=1, mu=0, sigma=1)

Arguments





alpha the value of the first shape parameter, must be positive, the default is 1

a the value of the second shape parameter, must be positive, the default is 1




Value


68 genpareto

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgenlogis4(x)pgenlogis4(x)vargenlogis4(x)esgenlogis4(x)

genpareto Generalized Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized Pareto distributiondue to Pickands (1975) given by

f(x) =1

k

(1− cx

k

)1/c−1

,

F (x) = 1−(

1− cx

k

)1/c

,

VaRp(X) =k

c[1− (1− p)c] ,

ESp(X) =k

c+k(1− p)c+1

pc(c+ 1)− k

pc(c+ 1)

for x < k/c if c > 0, x > k/c if c < 0, x > 0 if c = 0, 0 0, the scale parameter and−∞ < c <∞, the shape parameter.

Usage

dgenpareto(x, k=1, c=1, log=FALSE)pgenpareto(x, k=1, c=1, log.p=FALSE, lower.tail=TRUE)vargenpareto(p, k=1, c=1, log.p=FALSE, lower.tail=TRUE)esgenpareto(p, k=1, c=1)

Arguments



genpowerweibull 69

k the value of the scale parameter, must be positive, the default is 1

c the value of the shape parameter, can take any real value, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgenpareto(x)pgenpareto(x)vargenpareto(x)esgenpareto(x)

genpowerweibull Generalized power Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized power Weibull dis-tribution due to Nikulin and Haghighi (2006) given by

f(x) = aθxa−1 [1 + xa]θ−1

exp{

1− [1 + xa]θ},

F (x) = 1− exp{

1− [1 + xa]θ},

VaRp(X) ={

[1− log(1− p)]1/θ − 1}1/a

,

ESp(X) =1

p

∫ p

0

{[1− log(1− v)]

1/θ − 1}1/a

dv

for x > 0, 0 0, the first shape parameter, and θ > 0, the second shape parameter.

70 genpowerweibull

Usage

dgenpowerweibull(x, a=1, theta=1, log=FALSE)pgenpowerweibull(x, a=1, theta=1, log.p=FALSE, lower.tail=TRUE)vargenpowerweibull(p, a=1, theta=1, log.p=FALSE, lower.tail=TRUE)esgenpowerweibull(p, a=1, theta=1)

Arguments




theta the value of the second shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgenpowerweibull(x)pgenpowerweibull(x)vargenpowerweibull(x)esgenpowerweibull(x)

genunif 71

genunif Generalized uniform distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized uniform distributiongiven by

f(x) = hkc(x− a)c−1 [1− k(x− a)c]h−1

,

F (x) = 1− [1− k(x− a)c]h,

VaRp(X) = a+ k−1/c[1− (1− p)1/h

]1/c,

ESp(X) = a+k−1/c

p

∫ p

0

[1− (1− v)1/h

]1/cdv

for a ≤ x ≤ a + k−1/c, 0 0, the first shapeparameter, k > 0, the scale parameter, and h > 0, the second shape parameter.

Usage

dgenunif(x, a=0, c=1, h=1, k=1, log=FALSE)pgenunif(x, a=0, c=1, h=1, k=1, log.p=FALSE, lower.tail=TRUE)vargenunif(p, a=0, c=1, h=1, k=1, log.p=FALSE, lower.tail=TRUE)esgenunif(p, a=0, c=1, h=1, k=1)

Arguments



a the value of the location parameter, can take any real value, the default is zero

k the value of the scale parameter, must be positive, the default is 1

c the value of the first scale parameter, must be positive, the default is 1

h the value of the second scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

72 gev

References


Examples

x=runif(10,min=0,max=1)dgenunif(x)pgenunif(x)vargenunif(x)esgenunif(x)

gev Generalized extreme value distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized extreme value distri-bution due to Fisher and Tippett (1928) given by

f(x) =1

σ

[1 + ξ

(x− µσ

)]−1/ξ−1

exp

{−[1 + ξ

(x− µσ

)]−1/ξ},

F (x) = exp

{−[1 + ξ

(x− µσ

)]−1/ξ},

VaRp(X) = µ− σ

ξ+σ

ξ(− log p)−ξ,

ESp(X) = µ− σ

ξ+

σ

pξ

∫ p

0

(− log v)−ξdv

for x ≥ µ − σ/ξ if ξ > 0, x ≤ µ − σ/ξ if ξ < 0, −∞ < x < ∞ if ξ = 0, 0 0, the scale parameter, and −∞ < ξ < ∞, the shapeparameter.

Usage

dgev(x, mu=0, sigma=1, xi=1, log=FALSE)pgev(x, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)vargev(p, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)esgev(p, mu=0, sigma=1, xi=1)

Arguments




gompertz 73


xi the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgev(x)pgev(x)vargev(x)esgev(x)

gompertz Gompertz distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gompertz distribution due toGompertz (1825) given by

f(x) = bη exp(bx) exp [η − η exp(bx)] ,F (x) = 1− exp [η − η exp(bx)] ,

VaRp(X) =1

blog

[1− 1

ηlog(1− p)

],

ESp(X) =1

pb

∫ p

0

log

[1− 1

ηlog(1− v)

]dv

for x > 0, 0 0, the first scale parameter and η > 0, the second scale parameter.

74 gompertz

Usage

dgompertz(x, b=1, eta=1, log=FALSE)pgompertz(x, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)vargompertz(p, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)esgompertz(p, b=1, eta=1)

Arguments



b the value of the first scale parameter, must be positive, the default is 1

eta the value of the second scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgompertz(x)pgompertz(x)vargompertz(x)esgompertz(x)

gumbel 75

gumbel Gumbel distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gumbel distribution given by dueto Gumbel (1954) given by

f(x) =1

σexp

(µ− xσ

)exp

[− exp

(µ− xσ

)],

F (x) = exp

[− exp

(µ− xσ

)],

VaRp(X) = µ− σ log(− log p),

ESp(X) = µ− σ

p

∫ p

0

log(− log v)dv


Usage

dgumbel(x, mu=0, sigma=1, log=FALSE)pgumbel(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)vargumbel(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esgumbel(p, mu=0, sigma=1)

Arguments








Value


Author(s)

Saralees Nadarajah

76 gumbel2

References


Examples

x=runif(10,min=0,max=1)dgumbel(x)pgumbel(x)vargumbel(x)esgumbel(x)

gumbel2 Gumbel II distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gumbel II distribution

f(x) = abx−a−1 exp(−bx−a

),

F (x) = 1− exp(−bx−a

),

VaRp(X) = b1/a [− log(1− p)]−1/a,

ESp(X) =b1/a

p

∫ p

0

[− log(1− v)]−1/a

dv

for x > 0, 0 0, the shape parameter, and b > 0, the scale parameter.

Usage

dgumbel2(x, a=1, b=1, log=FALSE)pgumbel2(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)vargumbel2(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esgumbel2(p, a=1, b=1)

Arguments








halfcauchy 77

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dgumbel2(x)pgumbel2(x)vargumbel2(x)#esgumbel2(x)

halfcauchy Half Cauchy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the half Cauchy distribution given by

f(x) =2

π

σ

x2 + σ2,

F (x) =2

πarctan

(xσ

),

VaRp(X) = σ tan(πp

2

),

ESp(X) =σ

p

∫ p

0

tan(πv

2

)dv

for x > 0, 0 0, the scale parameter.

Usage

dhalfcauchy(x, sigma=1, log=FALSE)phalfcauchy(x, sigma=1, log.p=FALSE, lower.tail=TRUE)varhalfcauchy(p, sigma=1, log.p=FALSE, lower.tail=TRUE)eshalfcauchy(p, sigma=1)

78 halflogis

Arguments







Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dhalfcauchy(x)phalfcauchy(x)varhalfcauchy(x)eshalfcauchy(x)

halflogis Half logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the half logistic distribution given by

f(x) =2λ exp (−λx)

[1 + exp (−λx)]2 ,

F (x) =1− exp (−λx)

1 + exp (−λx),

VaRp(X) = − 1

λlog

1− p1 + p

,

ESp(X) = − 1

λlog

1− p1 + p

+1

λplog(1− p2

)for x > 0, 0 0, the scale parameter.

halflogis 79

Usage

dhalflogis(x, lambda=1, log=FALSE)phalflogis(x, lambda=1, log.p=FALSE, lower.tail=TRUE)varhalflogis(p, lambda=1, log.p=FALSE, lower.tail=TRUE)eshalflogis(p, lambda=1)

Arguments







Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dhalflogis(x)phalflogis(x)varhalflogis(x)eshalflogis(x)

80 halfnorm

halfnorm Half normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for Half normal distribution given by

f(x) =2

σφ(xσ

),

F (x) = 2Φ(xσ

)− 1,

VaRp(X) = σΦ−1

(1 + p

2

),

ESp(X) =σ

p

∫ p

0

Φ−1

(1 + v

2

)dv

for x > 0, 0 0, the scale parameter.

Usage

dhalfnorm(x, sigma=1, log=FALSE)phalfnorm(x, sigma=1, log.p=FALSE, lower.tail=TRUE)varhalfnorm(p, sigma=1, log.p=FALSE, lower.tail=TRUE)eshalfnorm(p, sigma=1)

Arguments







Value


Author(s)

Saralees Nadarajah

References


halfT 81

Examples

x=runif(10,min=0,max=1)dhalfnorm(x)phalfnorm(x)varhalfnorm(x)eshalfnorm(x)

halfT Half Student’s t distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the half Student’s t distribution givenby

f(x) =2Γ(n+1

2

)√nπΓ

(n2

) (1 +x2

n

)−n+12

,

F (x) = I x2

x2+n

(1

2,n

2

),

VaRp(X) =

√nI−1p

(12 ,

n2

)1− I−1

p

(12 ,

n2

) ,ESp(X) =

√n

p

∫ p

0

√I−1v

(12 ,

n2

)1− I−1

v

(12 ,

n2

)dvfor −∞ < x <∞, 0 0, the degree of freedom parameter.

Usage

dhalfT(x, n=1, log=FALSE)phalfT(x, n=1, log.p=FALSE, lower.tail=TRUE)varhalfT(p, n=1, log.p=FALSE, lower.tail=TRUE)eshalfT(p, n=1)

Arguments


to be computedn the value of the degree of freedom parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


82 HBlaplace

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dhalfT(x)phalfT(x)varhalfT(x)eshalfT(x)

HBlaplace Holla-Bhattacharya Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Holla-Bhattacharya Laplacedistribution due to Holla and Bhattacharya (1968) given by

f(x) =

aφ exp {φ (x− θ)} , if x ≤ θ,

(1− a)φ exp {φ (θ − x)} , if x > θ,

F (x) =

a exp (φx− θφ) , if x ≤ θ,

1− (1− a) exp (θφ− φx) , if x > θ,

VaRp(X) =

θ +

1

φlog(pa

), if p ≤ a,

θ − 1

φlog

(1− p1− a

), if p > a,

ESp(X) =

θ − 1

φ+

1

φlog

p

a, if p ≤ a,

1

p

[θ(1 + p− a) +

p− 2a− (1− a) log a

φ+

1− pφ

log1− p1− a

], if p > a

for −∞ < x < ∞, 0 0, the second scale parameter.

Usage

dHBlaplace(x, a=0.5, theta=0, phi=1, log=FALSE)pHBlaplace(x, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)varHBlaplace(p, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)esHBlaplace(p, a=0.5, theta=0, phi=1)

HL 83

Arguments


to be computedtheta the value of the location parameter, can take any real value, the default is zeroa the value of the first scale parameter, must be in the unit interval, the default is

0.5phi the value of the second scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dHBlaplace(x)pHBlaplace(x)varHBlaplace(x)esHBlaplace(x)

HL Hankin-Lee distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Hankin-Lee distribution due toHankin and Lee (2006) given by

VaRp(X) =cpa

(1− p)b,

ESp(X) =c

pBp(a+ 1, 1− b)

for 0 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the secondshape parameter.

84 Hlogis

Usage

varHL(p, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)esHL(p, a=1, b=1, c=1)

Arguments


c the value of the scale parameter, must be positive, the default is 1






Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)varHL(x)esHL(x)

Hlogis Hosking logistic distribution

Hlogis 85

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Hosking logistic distribution dueto Hosking (1989, 1990) given by

f(x) =(1− kx)1/k−1[

1 + (1− kx)1/k]2 ,

F (x) =1

1 + (1− kx)1/k,

VaRp(X) =1

k

[1−

(1− pp

)k],

ESp(X) =1

k− 1

kpBp(1− k, 1 + k)

for x < 1/k if k > 0, x > 1/k if k < 0, −∞ < x < ∞ if k = 0, and −∞ < k < ∞, the shapeparameter.

Usage

dHlogis(x, k=1, log=FALSE)pHlogis(x, k=1, log.p=FALSE, lower.tail=TRUE)varHlogis(p, k=1, log.p=FALSE, lower.tail=TRUE)esHlogis(p, k=1)

Arguments



k the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


86 invbeta

Examples

x=runif(10,min=0,max=1)dHlogis(x)pHlogis(x)varHlogis(x)esHlogis(x)

invbeta Inverse beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the inverse beta distribution given by

f(x) =xa−1

B(a, b)(1 + x)a+b,

F (x) = I x1+x

(a, b),

VaRp(X) =I−1p (a, b)

1− I−1p (a, b)

,

ESp(X) =1

p

∫ p

0

I−1v (a, b)

1− I−1v (a, b)

dv

for x > 0, 0 0, the first shape parameter, and b > 0, the second shape parameter.

Usage

dinvbeta(x, a=1, b=1, log=FALSE)pinvbeta(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varinvbeta(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esinvbeta(p, a=1, b=1)

Arguments








Value


invexpexp 87

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dinvbeta(x)pinvbeta(x)varinvbeta(x)esinvbeta(x)

invexpexp Inverse exponentiated exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the inverse exponentiated exponentialdistribution due to Ghitany et al. (2013) given by

f(x) = aλx−2 exp

(−λx

)[1− exp

(−λx

)]a−1

,

F (x) = 1−[1− exp

(−λx

)]a,

VaRp(X) = λ{− log

[1− (1− p)1/a

]}−1

,

ESp(X) =λ

p

∫ p

0

{− log

[1− (1− v)1/a

]}−1

dv


Usage

dinvexpexp(x, lambda=1, a=1, log=FALSE)pinvexpexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varinvexpexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esinvexpexp(p, lambda=1, a=1)

Arguments




88 invgamma

a the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dinvexpexp(x)pinvexpexp(x)varinvexpexp(x)esinvexpexp(x)

invgamma Inverse gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the inverse gamma distribution givenby

f(x) =ba exp(−b/x)

xa+1Γ(a),

F (x) = Q(a, b/x),

VaRp(X) = b[Q−1(a, p)

]−1,

ESp(X) =b

p

∫ p

0

[Q−1(a, v)

]−1dv

for x > 0, 0 0, the shape parameter, and b > 0, the scale parameter.

Usage

dinvgamma(x, a=1, b=1, log=FALSE)pinvgamma(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varinvgamma(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esinvgamma(p, a=1, b=1)

kum 89

Arguments


to be computedb the value of the scale parameter, must be positive, the default is 1a the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dinvgamma(x)pinvgamma(x)varinvgamma(x)esinvgamma(x)

kum Kumaraswamy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy distribution dueto Kumaraswamy (1980) given by

f(x) = abxa−1 (1− xa)b−1

,

F (x) = 1− (1− xa)b,

VaRp(X) =[1− (1− p)1/b

]1/a,

ESp(X) =1

p

∫ p

0

[1− (1− v)1/b

]1/adv

for 0 < x < 1, 0 0, the first shape parameter, and b > 0, the second shape parameter.

90 kum

Usage

dkum(x, a=1, b=1, log=FALSE)pkum(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkum(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskum(p, a=1, b=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkum(x)pkum(x)varkum(x)eskum(x)

kumburr7 91

kumburr7 Kumaraswamy Burr XII distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Burr XII distri-bution due to Parana\’iba et al. (2013) given by

f(x) =abkcxc−1

(1 + xc)k+1

[1− (1 + xc)

−k]a−1 {

1−[1− (1 + xc)

−k]a}b−1

,

F (x) = 1−{

1−[1− (1 + xc)

−k]a}b

,

VaRp(X) =

[{1−

[1− (1− p)1/b

]1/a}−1/k

− 1

]1/c

,

ESp(X) =1

p

∫ p

0

[{1−

[1− (1− v)1/b

]1/a}−1/k

− 1

]1/c

dv

for x > 0, 0 0, the first shape parameter, b > 0, the second shape parameter, c > 0,the third shape parameter, and k > 0, the fourth shape parameter.

Usage

dkumburr7(x, a=1, b=1, k=1, c=1, log=FALSE)pkumburr7(x, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)varkumburr7(p, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)eskumburr7(p, a=1, b=1, k=1, c=1)

Arguments






k the value of the fourth shape parameter, must be positive, the default is 1




Value


92 kumexp

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumburr7(x)pkumburr7(x)varkumburr7(x)eskumburr7(x)

kumexp Kumaraswamy exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy exponential dis-tribution due to Cordeiro and de Castro (2011) given by

f(x) = abλ exp(−λx) [1− exp(−λx)]a−1 {1− [1− exp(−λx)]

a}b−1,

F (x) = 1− {1− [1− exp(−λx)]a}b ,

VaRp(X) = − 1

λlog

{1−

[1− (1− p)1/b

]1/a},

ESp(X) = − 1

pλ

∫ p

0

log

{1−

[1− (1− v)1/b

]1/a}dv

for x > 0, 0 0, the first shape parameter, b > 0, the second shape parameter, andλ > 0, the scale parameter.

Usage

dkumexp(x, lambda=1, a=1, b=1, log=FALSE)pkumexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkumexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskumexp(p, lambda=1, a=1, b=1)

Arguments




kumgamma 93






Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumexp(x)pkumexp(x)varkumexp(x)eskumexp(x)

kumgamma Kumaraswamy gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy gamma distribu-tion due to de Pascoa et al. (2011) given by

f(x) = cdbaxa−1 exp(−bx)γc−1(a, bx)

Γc(a)

[1− γc(a, bx)

Γc(a)

]d−1

,

F (x) = 1−[1− γc(a, bx)

Γc(a)

]d,

VaRp(X) =1

bQ−1

(a, 1−

[1− (1− p)1/d

]1/c),

ESp(X) =1

bp

∫ p

0

Q−1

(a, 1−

[1− (1− v)1/d

]1/c)dv

for x > 0, 0 0, the first shape parameter, b > 0, the scale parameter, c > 0, thesecond shape parameter, and d > 0, the third shape parameter.

94 kumgamma

Usage

dkumgamma(x, a=1, b=1, c=1, d=1, log=FALSE)pkumgamma(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)varkumgamma(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)eskumgamma(p, a=1, b=1, c=1, d=1)

Arguments










Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumgamma(x)pkumgamma(x)varkumgamma(x)eskumgamma(x)

kumgumbel 95

kumgumbel Kumaraswamy Gumbel distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Gumbel distribu-tion due to Cordeiro et al. (2012a) given by

f(x) =ab

σexp

(µ− xσ

)exp

[−a exp

(µ− xσ

)]{1− exp

[−a exp

(µ− xσ

)]}b−1

,

F (x) = 1−{

1− exp

[−a exp

(µ− xσ

)]}b,

VaRp(X) = µ− σ log

{− log

[1− (1− p)1/b

]1/a},

ESp(X) = µ− σ

p

∫ p

0

log

{− log

[1− (1− v)1/b

]1/a}dv


Usage

dkumgumbel(x, a=1, b=1, mu=0, sigma=1, log=FALSE)pkumgumbel(x, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varkumgumbel(p, a=1, b=1, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eskumgumbel(p, a=1, b=1, mu=0, sigma=1)

Arguments










Value


96 kumhalfnorm

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumgumbel(x)pkumgumbel(x)varkumgumbel(x)eskumgumbel(x)

kumhalfnorm Kumaraswamy half normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy half normaldistribution due to Cordeiro et al. (2012c) given by

f(x) =2ab

σφ(xσ

) [2Φ(xσ

)− 1]a−1 {

1−[2Φ(xσ

)− 1]a}b−1

,

F (x) = 1−{

1−[2Φ(xσ

)− 1]a}b

,

VaRp(X) = σΦ−1

(1

2+

1

2

[1− (1− p)1/b

]1/a),

ESp(X) =σ

p

∫ p

0

Φ−1

(1

2+

1

2

[1− (1− v)1/b

]1/a)dv

for x > 0, 0 0, the scale parameter, a > 0, the first shape parameter, and b > 0, thesecond shape parameter.

Usage

dkumhalfnorm(x, sigma=1, a=1, b=1, log=FALSE)pkumhalfnorm(x, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkumhalfnorm(p, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskumhalfnorm(p, sigma=1, a=1, b=1)

kumloglogis 97

Arguments









Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumhalfnorm(x)pkumhalfnorm(x)varkumhalfnorm(x)eskumhalfnorm(x)

kumloglogis Kumaraswamy log-logistic distribution

98 kumloglogis

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy log-logistic dis-tribution due to de Santana et al. (2012) given by

f(x) =abβαβxaβ−1

(αβ + xβ)a+1

[1− xaβ

(αβ + xβ)a

]b−1

,

F (x) =

[1− xaβ

(αβ + xβ)a

]b,

VaRp(X) = α

{[1− (1− p)1/b

]1/a− 1

}−1/β

,

ESp(X) =α

p

∫ p

0

{[1− (1− v)1/b

]1/a− 1

}−1/β

dv

for x > 0, 0 0, the scale parameter, β > 0, the first shape parameter, a > 0, thesecond shape parameter, and b > 0, the third shape parameter.

Usage

dkumloglogis(x, a=1, b=1, alpha=1, beta=1, log=FALSE)pkumloglogis(x, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)varkumloglogis(p, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)eskumloglogis(p, a=1, b=1, alpha=1, beta=1)

Arguments



alpha the value of the scale parameter, must be positive, the default is 1

beta the value of the first shape parameter, must be positive, the default is 1

a the value of the second shape parameter, must be positive, the default is 1

b the value of the third shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

kumnormal 99

References


Examples

x=runif(10,min=0,max=1)dkumloglogis(x)pkumloglogis(x)varkumloglogis(x)eskumloglogis(x)

kumnormal Kumaraswamy normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for Kumaraswamy normal distributiondue to Cordeiro and de Castro (2011) given by

f(x) =ab

σφ

(x− µσ

)Φa−1

(x− µσ

)[1− Φa

(x− µσ

)]b−1

,

F (x) = 1−[1− Φa

(x− µσ

)]b,

VaRp(X) = µ+ σΦ−1

([1− (1− p)1/b

]1/a),

ESp(X) = µ+σ

p

∫ p

0

Φ−1

([1− (1− v)1/b

]1/a)dv


Usage

dkumnormal(x, mu=0, sigma=1, a=1, b=1, log=FALSE)pkumnormal(x, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varkumnormal(p, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eskumnormal(p, mu=0, sigma=1, a=1, b=1)

Arguments





100 kumpareto






Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumnormal(x)pkumnormal(x)varkumnormal(x)eskumnormal(x)

kumpareto Kumaraswamy Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Pareto distributiondue to Pereira et al. (2013) given by

f(x) = abcKcx−c−1

[1−

(K

x

)c]a−1{1−

[1−

(K

x

)c]a}b−1

,

F (x) = 1−{

1−[1−

(K

x

)c]a}b,

VaRp(X) = K

{1−

[1− (1− p)1/b

]1/a}−1/c

,

ESp(X) =K

p

∫ p

0

{1−

[1− (1− v)1/b

]1/a}−1/c

dv

for x ≥ K, 0 0, the scale parameter, c > 0, the first shape parameter, a > 0, thesecond shape parameter, and b > 0, the third shape parameter.

kumpareto 101

Usage

dkumpareto(x, K=1, a=1, b=1, c=1, log=FALSE)pkumpareto(x, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)varkumpareto(p, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)eskumpareto(p, K=1, a=1, b=1, c=1)

Arguments



K the value of the scale parameter, must be positive, the default is 1







Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumpareto(x)pkumpareto(x)varkumpareto(x)eskumpareto(x)

102 kumweibull

kumweibull Kumaraswamy Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Weibull distribu-tion due to Cordeiro et al. (2010) given by

f(x) =abαxα−1

σαexp

[−(xσ

)α]{1− exp

[−(xσ

)α]}a−1[1−

{1− exp

[−(xσ

)α]}a]b−1

,

F (x) = 1−[1−

{1− exp

[−(xσ

)α]}a]b,

VaRp(X) = σ

[− log

{1−

[1− (1− p)1/b

]1/a}]1/α

,

ESp(X) =σ

p

∫ p

0

[− log

{1−

[1− (1− v)1/b

]1/a}]1/α

dv

for x > 0, 0 0, the first shape parameter, b > 0, the second shape parameter, α > 0,the third shape parameter, and σ > 0, the scale parameter.

Usage

dkumweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)pkumweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varkumweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)eskumweibull(p, a=1, b=1, alpha=1, sigma=1)

Arguments










Value


laplace 103

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dkumweibull(x)pkumweibull(x)varkumweibull(x)eskumweibull(x)

laplace Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Laplace distribution due to dueto Laplace (1774) given by

f(x) =1

2σexp

(−| x− µ |

σ

),

F (x) =

1

2exp

(x− µσ

), if x < µ,

1− 1

2exp

(−x− µ

σ

), if x ≥ µ,

VaRp(X) =

µ+ σ log(2p), if p < 1/2,

µ− σ log [2(1− p)] , if p ≥ 1/2,

ESp(X) =

µ+ σ [log(2p)− 1] , if p < 1/2,

µ+ σ − σ

p+ σ

1− pp

log(1− p) + σ1− pp

log 2, if p ≥ 1/2


Usage

dlaplace(x, mu=0, sigma=1, log=FALSE)plaplace(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlaplace(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslaplace(p, mu=0, sigma=1)

104 lfr

Arguments


to be computedmu the value of the location parameter, can take any real value, the default is zerosigma the value of the scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dlaplace(x)plaplace(x)varlaplace(x)eslaplace(x)

lfr Linear failure rate distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the linear failure rate distribution dueto Bain (1974) given by

f(x) = (a+ bx) exp(−ax− bx2/2

),

F (x) = 1− exp(−ax− bx2/2

),

VaRp(X) =−a+

√a2 − 2b log(1− p)

b,

ESp(X) = −ab

+1

bp

∫ p

0

√a2 − 2b log(1− v)dv

for x > 0, 0 0, the first scale parameter, and b > 0, the second scale parameter.

lfr 105

Usage

dlfr(x, a=1, b=1, log=FALSE)plfr(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varlfr(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)eslfr(p, a=1, b=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dlfr(x)plfr(x)varlfr(x)eslfr(x)

106 LNbeta

LNbeta Libby-Novick beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Libby-Novick beta distributiondue to Libby and Novick (1982) given by

f(x) =λaxa−1(1− x)b−1

B(a, b) [1− (1− λ)x]a+b

,

F (x) = I λx1+(λ−1)x

(a, b),

VaRp(X) =I−1p (a, b)

λ− (λ− 1)I−1p (a, b)

,

ESp(X) =1

p

∫ p

0

I−1v (a, b)

λ− (λ− 1)I−1v (a, b)

dv

for 0 < x < 1, 0 0, the scale parameter, a > 0, the first shape parameter, and b > 0,the second shape parameter.

Usage

dLNbeta(x, lambda=1, a=1, b=1, log=FALSE)pLNbeta(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varLNbeta(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esLNbeta(p, lambda=1, a=1, b=1)

Arguments









Value


Author(s)

Saralees Nadarajah

logbeta 107

References


Examples

x=runif(10,min=0,max=1)dLNbeta(x)pLNbeta(x)varLNbeta(x)esLNbeta(x)

logbeta Log beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log beta distribution given by

f(x) =(log d− log c)1−a−b

xB(a, b)(log x− log c)a−1(log d− log x)b−1,

F (x) = I log x−log clog d−log c

(a, b),

VaRp(X) = c

(d

c

)I−1p (a,b)

,

ESp(X) =c

p

∫ p

0

(d

c

)I−1v (a,b)

dv

for 0 < c ≤ x ≤ d, 0 0, the first shape parameter, b > 0, the second shape parameter,c > 0, the first location parameter, and d > 0, the second location parameter.

Usage

dlogbeta(x, a=1, b=1, c=1, d=2, log=FALSE)plogbeta(x, a=1, b=1, c=1, d=2, log.p=FALSE, lower.tail=TRUE)varlogbeta(p, a=1, b=1, c=1, d=2, log.p=FALSE, lower.tail=TRUE)eslogbeta(p, a=1, b=1, c=1, d=2)

Arguments



c the value of the first location parameter, must be positive, the default is 1

d the value of the second location parameter, must be positive and greater than c,the default is 2


108 logcauchy

b the value of the second scale parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dlogbeta(x)plogbeta(x)varlogbeta(x)eslogbeta(x)

logcauchy Log Cauchy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log Cauchy distribution given by

f(x) =1

xπ

σ

(log x− µ)2 + σ2,

F (x) =1

πarctan

(log x− µ

σ

),

VaRp(X) = exp [µ+ σ tan (πp)] ,

ESp(X) =exp(µ)

p

∫ p

0

exp [σ tan (πv)] dv

for x > 0, 0 0, the scale parameter.

Usage

dlogcauchy(x, mu=0, sigma=1, log=FALSE)plogcauchy(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlogcauchy(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslogcauchy(p, mu=0, sigma=1)

loggamma 109

Arguments



Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dlogcauchy(x)plogcauchy(x)varlogcauchy(x)#eslogcauchy(x)

loggamma Log gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log gamma distribution due toConsul and Jain (1971) given by

f(x) =arxa−1(− log x)r−1

Γ(r),

F (x) = Q(r,−a log x),

VaRp(X) = exp

[−1

aQ−1(r, p)

],

ESp(X) =1

p

∫ p

0

exp

[−1

aQ−1(r, v)

]dv

for x > 0, 0 0, the first shape parameter, and r > 0, the second shape parameter.

110 loggamma

Usage

dloggamma(x, a=1, r=1, log=FALSE)ploggamma(x, a=1, r=1, log.p=FALSE, lower.tail=TRUE)varloggamma(p, a=1, r=1, log.p=FALSE, lower.tail=TRUE)esloggamma(p, a=1, r=1)

Arguments




r the value of the second scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dloggamma(x)ploggamma(x)varloggamma(x)esloggamma(x)

logisexp 111

logisexp Logistic exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic exponential distributiondue to Lan and Leemis (2008) given by

f(x) =aλ exp(λx) [exp(λx)− 1]

a−1

{1 + [exp(λx)− 1]a}2

,

F (x) =[exp(λx)− 1]

a

1 + [exp(λx)− 1]a ,

VaRp(X) =1

λlog

[1 +

(p

1− p

)1/a],

ESp(X) =1

pλ

∫ p

0

log

[1 +

(v

1− v

)1/a]dv


Usage

dlogisexp(x, lambda=1, a=1, log=FALSE)plogisexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varlogisexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)eslogisexp(p, lambda=1, a=1)

Arguments








Value


Author(s)

Saralees Nadarajah

112 logisrayleigh

References


Examples

x=runif(10,min=0,max=1)dlogisexp(x)plogisexp(x)varlogisexp(x)eslogisexp(x)

logisrayleigh Logistic Rayleigh distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic Rayleigh distribution dueto Lan and Leemis (2008) given by

f(x) = aλx exp(λx2/2

) [exp

(λx2/2

)− 1]a−1

{1 +

[exp

(λx2/2

)− 1]a}−2

,

F (x) =

[exp

(λx2/2

)− 1]a

1 + [exp (λx2/2)− 1]a ,

VaRp(X) =

√2

λ

√√√√log

[1 +

(p

1− p

)1/a],

ESp(X) =

√2

p√λ

∫ p

0

{log

[1 +

(v

1− v

)1/a]}1/2

dv

for x > 0, 0 0, the shape parameter, and λ > 0, the scale parameter.

Usage

dlogisrayleigh(x, a=1, lambda=1, log=FALSE)plogisrayleigh(x, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varlogisrayleigh(p, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)eslogisrayleigh(p, a=1, lambda=1)

Arguments





logistic 113

log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dlogisrayleigh(x)plogisrayleigh(x)varlogisrayleigh(x)eslogisrayleigh(x)

logistic Logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic distribution given by

f(x) =1

σexp

(−x− µ

σ

)[1 + exp

(−x− µ

σ

)]−2

,

F (x) =1

1 + exp(−x−µσ

) ,VaRp(X) = µ+ σ log [p(1− p)] ,ESp(X) = µ− 2σ + σ log p− σ 1− p

plog(1− p)


Usage

dlogistic(x, mu=0, sigma=1, log=FALSE)plogistic(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlogistic(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslogistic(p, mu=0, sigma=1)

114 loglaplace

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dlogistic(x)plogistic(x)varlogistic(x)eslogistic(x)

loglaplace Log Laplace distribution

loglaplace 115

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log Laplace distribution given by

f(x) =

abxb−1

δb(a+ b), if x ≤ δ,

abδa

xa+1(a+ b), if x > δ,

F (x) =

axb

δb(a+ b), if x ≤ δ,

1− bδa

xa(a+ b), if x > δ,

VaRp(X) =

δ

[pa+ b

a

]1/b

, if p ≤ aa+b ,

δ

[(1− p)a+ b

a

]−1/a

, if p > aa+b ,

ESp(X) =

δb

b+ 1

[pa+ b

a

]1/b

, if p ≤ aa+b ,

aδ

p(1 + 1/b)(a+ b)+

a1/ab1−1/aδ

p(a+ b)(1− 1/a)

− δ(1− p)p(1− 1/a)

[a

(a+ b)(1− p)

]1/a

, if p > aa+b

for −∞ < x < ∞, 0 0, the scale parameter, a > 0, the first shape parameter, andb > 0, the second shape parameter.

Usage

dloglaplace(x, a=1, b=1, delta=0, log=FALSE)ploglaplace(x, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)varloglaplace(p, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)esloglaplace(p, a=1, b=1, delta=0)

Arguments


to be computeddelta the value of the scale parameter, must be positive, the default is 1a the value of the first shape parameter, must be positive, the default is 1b the value of the second shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

116 loglog

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dloglaplace(x)ploglaplace(x)varloglaplace(x)esloglaplace(x)

loglog Loglog distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Loglog distribution due to Pham(2002) given by

f(x) = a log(λ)xa−1λxa

exp[1− λx

a],

F (x) = 1− exp[1− λx

a],

VaRp(X) =

{log [1− log(1− p)]

log λ

}1/a

,

ESp(X) =1

p(log λ)1/a

∫ p

0

{log [1− log(1− v)]}1/a dv

for x > 0, 0 0, the shape parameter, and λ > 1, the scale parameter.

Usage

dloglog(x, a=1, lambda=2, log=FALSE)ploglog(x, a=1, lambda=2, log.p=FALSE, lower.tail=TRUE)varloglog(p, a=1, lambda=2, log.p=FALSE, lower.tail=TRUE)esloglog(p, a=1, lambda=2)

loglogis 117

Arguments



lambda the value of the scale parameter, must be greater than 1, the default is 2





Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dloglog(x)ploglog(x)varloglog(x)esloglog(x)

loglogis Log-logistic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log-logistic distribution given by

f(x) =babxb−1

(ab + xb)2 ,

F (x) =xb

ab + xb,

VaRp(X) = a

(p

1− p

)1/b

,

ESp(X) =a

pBp

(1 +

1

b, 1− 1

b

)

118 loglogis

for x > 0, 0 0, the scale parameter, and b > 0, the shape parameter, whereBx(a, b) =

∫ x0ta−1(1− t)b−1dt denotes the incomplete beta function.

Usage

dloglogis(x, a=1, b=1, log=FALSE)ploglogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varloglogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esloglogis(p, a=1, b=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dloglogis(x)ploglogis(x)varloglogis(x)esloglogis(x)

lognorm 119

lognorm Lognormal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the lognormal distribution given by

f(x) =1

σxφ

(log x− µ

σ

),

F (x) = Φ

(log x− µ

σ

),

VaRp(X) = exp[µ+ σΦ−1(p)

],

ESp(X) =exp(µ)

p

∫ p

0

exp[σΦ−1(v)

]dv

for x > 0, 0 0, the scale parameter.

Usage

dlognorm(x, mu=0, sigma=1, log=FALSE)plognorm(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varlognorm(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)eslognorm(p, mu=0, sigma=1)

Arguments








Value


Author(s)

Saralees Nadarajah

120 lomax

References


Examples

x=runif(10,min=0,max=1)dlognorm(x)plognorm(x)varlognorm(x)eslognorm(x)

lomax Lomax distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Lomax distribution due to Lomax(1954) given by

f(x) =a

λ

(1 +

x

λ

)−a−1

,

F (x) = 1−(

1 +x

λ

)−a,

VaRp(X) = λ[(1− p)−1/a − 1

],

ESp(X) = −λ+λ− λ(1− p)1−1/a

p− p/afor x > 0, 0 0, the shape parameter, and λ > 0, the scale parameter.

Usage

dlomax(x, a=1, lambda=1, log=FALSE)plomax(x, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varlomax(p, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)eslomax(p, a=1, lambda=1)

Arguments








Mlaplace 121

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dlomax(x)plomax(x)varlomax(x)eslomax(x)

Mlaplace McGill Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the McGill Laplace distribution dueto McGill (1962) given by

f(x) =

1

2ψexp

(x− θψ

), if x ≤ θ,

1

2φexp

(θ − xφ

), if x > θ,

F (x) =

1

2exp

(x− θψ

), if x ≤ θ,

1− 1

2exp

(θ − xφ

), if x > θ,

VaRp(X) =

θ + ψ log(2p), if p ≤ 1/2,

θ − φ log (2(1− p)) , if p > 1/2,

ESp(X) =

ψ + θ log(2p)− θp, if p ≤ 1/2,

θ + φ+ψ − φ− 2θ

2p+φ

plog 2− φ log 2

+φ

plog(1− p)− φ log(1− p), if p > 1/2

122 Mlaplace

for −∞ < x < ∞, 0 0, the first scaleparameter, and ψ > 0, the second scale parameter.

Usage

dMlaplace(x, theta=0, phi=1, psi=1, log=FALSE)pMlaplace(x, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)varMlaplace(p, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)esMlaplace(p, theta=0, phi=1, psi=1)

Arguments




phi the value of the first scale parameter, must be positive, the default is 1

psi the value of the second scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dMlaplace(x)pMlaplace(x)varMlaplace(x)esMlaplace(x)

moexp 123

moexp Marshall-Olkin exponential distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin exponentialdistribution due to Marshall and Olkin (1997) given by

f(x) =λ exp(λx)

[exp(λx)− 1 + a]2 ,

F (x) =exp(λx)− 2 + a

exp(λx)− 1 + a,

VaRp(X) =1

λlog

2− a− (1− a)p

1− p,

ESp(X) =1

λlog [2− a− (1− a)p]− 2− a

λ(1− a)plog

2− a− (1− a)p

2− a+

1− pλp

log(1− p)

for x > 0, 0 0, the first scale parameter and λ > 0, the second scale parameter.

Usage

dmoexp(x, lambda=1, a=1, log=FALSE)pmoexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)varmoexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)esmoexp(p, lambda=1, a=1)

Arguments








Value


Author(s)

Saralees Nadarajah

124 moweibull

References


Examples

x=runif(10,min=0,max=1)dmoexp(x)pmoexp(x)varmoexp(x)esmoexp(x)

moweibull Marshall-Olkin Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin Weibull distribu-tion due to Marshall and Olkin (1997) given by

f(x) = bλbxb−1 exp[(λx)b

] {exp

[(λx)b

]− 1 + a

}−2,

F (x) =exp

[(λx)b

]− 2 + a

exp[(λx)b

]− 1 + a

,

VaRp(X) =1

λ

[log

(1

1− p+ 1− a

)]1/b

,

ESp(X) =1

λp

∫ p

0

[log

(1

1− v+ 1− a

)]1/b

dv

for x > 0, 0 0, the first scale parameter, b > 0, the shape parameter, and λ > 0, thesecond scale parameter.

Usage

dmoweibull(x, a=1, b=1, lambda=1, log=FALSE)pmoweibull(x, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varmoweibull(p, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esmoweibull(p, a=1, b=1, lambda=1)

Arguments






MRbeta 125




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dmoweibull(x)pmoweibull(x)varmoweibull(x)esmoweibull(x)

MRbeta McDonald-Richards beta distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the McDonald-Richards beta distri-bution due to McDonald and Richards (1987a, 1987b) given by

f(x) =xar−1 (bqr − xr)b−1

(bqr)a+b−1

B(a, b),

F (x) = I xrbqr

(a, b),

VaRp(X) = b1/rq[I−1p (a, b)

]1/r,

ESp(X) =b1/rq

p

∫ p

0

[I−1v (a, b)

]1/rdv

for 0 ≤ x ≤ b1/rq, 0 0, the scale parameter, a > 0, the first shape parameter, b > 0,the second shape parameter, and r > 0, the third shape parameter.

126 MRbeta

Usage

dMRbeta(x, a=1, b=1, r=1, q=1, log=FALSE)pMRbeta(x, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)varMRbeta(p, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)esMRbeta(p, a=1, b=1, r=1, q=1)

Arguments



q the value of the scale parameter, must be positive, the default is 1



r the value of the third shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dMRbeta(x)pMRbeta(x)varMRbeta(x)esMRbeta(x)

nakagami 127

nakagami Nakagami distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Nakagami distribution due toNakagami (1960) given by

f(x) =2mm

Γ(m)amx2m−1 exp

(−mx

2

a

),

F (x) = 1−Q(m,

mx2

a

),

VaRp(X) =

√a

m

√Q−1(m, 1− p),

ESp(X) =

√a

p√m

∫ p

0

√Q−1(m, 1− v)dv

for x > 0, 0 0, the scale parameter, and m > 0, the shape parameter.

Usage

dnakagami(x, m=1, a=1, log=FALSE)pnakagami(x, m=1, a=1, log.p=FALSE, lower.tail=TRUE)varnakagami(p, m=1, a=1, log.p=FALSE, lower.tail=TRUE)esnakagami(p, m=1, a=1)

Arguments




m the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

128 normal

References


Examples

x=runif(10,min=0,max=1)dnakagami(x)pnakagami(x)varnakagami(x)esnakagami(x)

normal Normal distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the normal distribution due to deMoivre (1738) and Gauss (1809) given by

f(x) =1

σφ

(x− µσ

),

F (x) = Φ

(x− µσ

),

VaRp(X) = µ+ σΦ−1(p),

ESp(X) = µ+σ

p

∫ p

0

Φ−1(v)dv

for −∞ < x < ∞, 0 0, the scaleparameter, where φ(·) denotes the pdf of a standard normal random variable, and Φ(·) denotes thecdf of a standard normal random variable.

Usage

dnormal(x, mu=0, sigma=1, log=FALSE)pnormal(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)varnormal(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)esnormal(p, mu=0, sigma=1)

Arguments



pareto 129

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dnormal(x)pnormal(x)varnormal(x)esnormal(x)

pareto Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Pareto distribution due to Pareto(1964) given by

f(x) = cKcx−c−1,

F (x) = 1−(K

x

)c,

VaRp(X) = K(1− p)−1/c,

ESp(X) =Kc

p(1− c)(1− p)1−1/c − Kc

p(1− c)

for x ≥ K, 0 0, the scale parameter, and c > 0, the shape parameter.

Usage

dpareto(x, K=1, c=1, log=FALSE)ppareto(x, K=1, c=1, log.p=FALSE, lower.tail=TRUE)varpareto(p, K=1, c=1, log.p=FALSE, lower.tail=TRUE)espareto(p, K=1, c=1)

130 paretostable

Arguments


to be computedK the value of the scale parameter, must be positive, the default is 1c the value of the shape parameter, must be positive, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dpareto(x)ppareto(x)varpareto(x)espareto(x)

paretostable Pareto positive stable distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Pareto positive stable distributiondue to Sarabia and Prieto (2009) and Guillen et al. (2011) given by

f(x) =νλ

x

[log(xσ

)]ν−1

exp{−λ[log(xσ

)]ν},

F (x) = 1− exp{−λ[log(xσ

)]ν},

VaRp(X) = σ exp

{[− 1

λlog(1− p)

]1/ν},

ESp(X) =σ

p

∫ p

0

exp

{[− 1

λlog(1− v)

]1/ν}dv

paretostable 131

for x > 0, 0 0, the first scale parameter, σ > 0, the second scale parameter, andν > 0, the shape parameter.

Usage

dparetostable(x, lambda=1, nu=1, sigma=1, log=FALSE)pparetostable(x, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varparetostable(p, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esparetostable(p, lambda=1, nu=1, sigma=1)

Arguments



lambda the value of the first scale parameter, must be positive, the default is 1

sigma the value of the second scale parameter, must be positive, the default is 1

nu the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dparetostable(x)pparetostable(x)varparetostable(x)esparetostable(x)

132 PCTAlaplace

PCTAlaplace Poiraud-Casanova-Thomas-Agnan Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Poiraud-Casanova-Thomas-Agnan Laplace distribution due to Poiraud-Casanova and Thomas-Agnan (2000) given by

f(x) =

a (1− a) exp {(1− a) (x− θ)} , if x ≤ θ,

a (1− a) exp {a (θ − x)} , if x > θ,

F (x) =

a exp {(1− a) (x− θ)} , if x ≤ θ,

1− (1− a) exp {a (θ − x)} , if x > θ,

VaRp(X) =

θ +

1

1− alog(pa

), if p ≤ a,

θ − 1

alog

(1− p1− a

), if p > a,

ESp(X) =

θ − log a

1− a+

log p− 1

(1− a)p, if p ≤ a,

θ − 1

a+

1

p− a

(1− a)p+

1− pap

log

(1− p1− a

), if p > a

for −∞ < x < ∞, 0 0, the scaleparameter.

Usage

dPCTAlaplace(x, a=0.5, theta=0, log=FALSE)pPCTAlaplace(x, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)varPCTAlaplace(p, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)esPCTAlaplace(p, a=0.5, theta=0)

Arguments




a the value of the scale parameter, must be in the unit interval, the default is 0.5




perks 133

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dPCTAlaplace(x)pPCTAlaplace(x)varPCTAlaplace(x)esPCTAlaplace(x)

perks Perks distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Perks distribution due to Perks(1932) given by

f(x) =a exp(bx) [1 + a]

[1 + a exp(bx)]2 ,

F (x) = 1− 1 + a

1 + a exp(bx),

VaRp(X) =1

blog

a+ p

a(1− p),

ESp(X) = −(

1 +a

p

)log a

b+

(a+ p) log(a+ p)

bp+

(1− p) log(1− p)bp

for x > 0, 0 0, the first scale parameter and b > 0, the second scale parameter.

Usage

dperks(x, a=1, b=1, log=FALSE)pperks(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)varperks(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)esperks(p, a=1, b=1)

134 power1

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dperks(x)pperks(x)varperks(x)esperks(x)

power1 Power function I distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the power function I distributiongiven by

f(x) = axa−1,F (x) = xa,

VaRp(X) = p1/a,

ESp(X) =p1/a

1/a+ 1

for 0 < x < 1, 0 0, the shape parameter.

power1 135

Usage

dpower1(x, a=1, log=FALSE)ppower1(x, a=1, log.p=FALSE, lower.tail=TRUE)varpower1(p, a=1, log.p=FALSE, lower.tail=TRUE)espower1(p, a=1)

Arguments







Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dpower1(x)ppower1(x)varpower1(x)espower1(x)

136 power2

power2 Power function II distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the power function II distributiongiven by

f(x) = b(1− x)b−1,

F (x) = 1− (1− x)b,

VaRp(X) = 1− (1− p)1/b,

ESp(X) = 1 +b[(1− p)1/b+1 − 1

]p(b+ 1)

for 0 < x < 1, 0 0, the shape parameter.

Usage

dpower2(x, b=1, log=FALSE)ppower2(x, b=1, log.p=FALSE, lower.tail=TRUE)varpower2(p, b=1, log.p=FALSE, lower.tail=TRUE)espower2(p, b=1)

Arguments







Value


Author(s)

Saralees Nadarajah

References


quad 137

Examples

x=runif(10,min=0,max=1)dpower2(x)ppower2(x)varpower2(x)espower2(x)

quad Quadratic distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the quadratic distribution given by

f(x) = α(x− β)2,

F (x) =α

3

[(x− β)3 + (β − a)3

],

VaRp(X) = β +

[3p

α− (β − a)3

]1/3

,

ESp(X) = β +α

4p

{[3p

α− (β − a)3

]4/3

− (β − a)4

}

for a ≤ x ≤ b, 0 < p < 1, −∞ < a < ∞ , the first location parameter, and −∞ < a < b < ∞,the second location parameter, where α = 12

(b−a)3 and β = a+b2 .

Usage

dquad(x, a=0, b=1, log=FALSE)pquad(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)varquad(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)esquad(p, a=0, b=1)

Arguments




b the value of the second location parameter, can take any real value but must begreater than a, the default is 1




138 rgamma

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dquad(x)pquad(x)varquad(x)esquad(x)

rgamma Reflected gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the reflected gamma distribution dueto Borgi (1965) given by

f(x) =1

2φΓ (a)

∣∣∣∣x− θφ∣∣∣∣a−1

exp

{−∣∣∣∣x− θφ

∣∣∣∣} ,F (x) =

1

2Q

(a,θ − xφ

), if x ≤ θ,

1− 1

2Q

(a,x− θφ

), if x > θ,

VaRp(X) =

θ − φQ−1 (a, 2p) , if p ≤ 1/2,

θ + φQ−1 (a, 2(1− p)) , if p > 1/2,

ESp(X) =

θ − φ

p

∫ p

0

Q−1 (a, 2v) dv, if p ≤ 1/2,

θ − φ

p

∫ 1/2

0

Q−1 (a, 2v) dv +φ

p

∫ p

1/2

Q−1 (a, 2(1− v)) dv, if p > 1/2

for −∞ < x <∞, 0 0, the scale parameter,and a > 0, the shape parameter.

rgamma 139

Usage

drgamma(x, a=1, theta=0, phi=1, log=FALSE)prgamma(x, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)varrgamma(p, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)esrgamma(p, a=1, theta=0, phi=1)

Arguments




phi the value of the scale parameter, must be positive, the default is 1





Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)drgamma(x)prgamma(x)varrgamma(x)esrgamma(x)

140 RS

RS Ramberg-Schmeiser distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Ramber-Schmeiser distributiondue to Ramberg and Schmeiser (1974) given by

VaRp(X) =pb − (1− p)c

d,

ESp(X) =pb

d(b+ 1)+

(1− p)c+1 − 1

pd(c+ 1)

for 0 0, the first shape parameter, c > 0, the second shape parameter, and d > 0, thescale parameter.

Usage

varRS(p, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)esRS(p, b=1, c=1, d=1)

Arguments


d the value of the scale parameter, must be positive, the default is 1






Value


Author(s)

Saralees Nadarajah

References


schabe 141

Examples

x=runif(10,min=0,max=1)varRS(x)esRS(x)

schabe Schabe distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Schabe distribution due to Schabe(1994) given by

f(x) =2γ + (1− γ)x/θ

θ(γ + x/θ)2 ,

F (x) =(1 + γ)x

x+ γθ,

VaRp(X) =pγθ

1 + γ − p,

ESp(X) = −θγ − θγ(1 + γ)

plog

1 + γ − p1 + γ

for x > 0, 0 0, the second scale parameter.

Usage

dschabe(x, gamma=0.5, theta=1, log=FALSE)pschabe(x, gamma=0.5, theta=1, log.p=FALSE, lower.tail=TRUE)varschabe(p, gamma=0.5, theta=1, log.p=FALSE, lower.tail=TRUE)esschabe(p, gamma=0.5, theta=1)

Arguments



gamma the value of the first scale parameter, must be in the unit interval, the default is0.5

theta the value of the second scale parameter, must be positive, the default is 1




Value


142 secant

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dschabe(x)pschabe(x)varschabe(x)esschabe(x)

secant Hyperbolic secant distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the hyperbolic secant distributiongiven by

f(x) =1

2sech

(πx2

),

F (x) =2

πarctan

[exp

(πx2

)],

VaRp(X) =2

πlog[tan

(πp2

)],

ESp(X) =2

πp

∫ p

0

log[tan

(πv2

)]dv

for −∞ < x <∞, and 0 < p < 1.

Usage

dsecant(x, log=FALSE)psecant(x, log.p=FALSE, lower.tail=TRUE)varsecant(p, log.p=FALSE, lower.tail=TRUE)essecant(p)

Arguments


to be computedlog if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

stacygamma 143

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dsecant(x)psecant(x)varsecant(x)essecant(x)

stacygamma Stacy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for Stacy distribution due to Stacy (1962)given by

f(x) =cxcγ−1 exp [−(x/θ)c]

θcγΓ(γ),

F (x) = 1−Q(γ,(xθ

)c),

VaRp(X) = θ[Q−1(γ, 1− p)

]1/c,

ESp(X) =θ

p

∫ p

0

[Q−1(γ, 1− v)

]1/cdv

for x > 0, 0 0, the scale parameter, c > 0, the first shape parameter, and γ > 0, thesecond shape parameter.

Usage

dstacygamma(x, gamma=1, c=1, theta=1, log=FALSE)pstacygamma(x, gamma=1, c=1, theta=1, log.p=FALSE, lower.tail=TRUE)varstacygamma(p, gamma=1, c=1, theta=1, log.p=FALSE, lower.tail=TRUE)esstacygamma(p, gamma=1, c=1, theta=1)

144 T

Arguments



theta the value of the scale parameter, must be positive, the default is 1

c the value of the first scale parameter, must be positive, the default is 1

gamma the value of the second scale parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dstacygamma(x)pstacygamma(x)varstacygamma(x)esstacygamma(x)

T Student’s t distribution

T 145

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Student’s t distribution due toGosset (1908) given by

f(x) =Γ(n+1

2

)√nπΓ

(n2

) (1 +x2

n

)−n+12

,

F (x) =1 + sign(x)

2− sign(x)

2I nx2+n

(n

2,

1

2

),

VaRp(X) =√nsign

(p− 1

2

)√1

I−1a

(n2 ,

12

) − 1,

where a = 2p if p < 1/2, a = 2(1− p) if p ≥ 1/2,

ESp(X) =

√n

p

∫ p

0

sign

(v − 1

2

)√1

I−1a

(n2 ,

12

) − 1dv,

where a = 2v if v < 1/2, a = 2(1− v) if v ≥ 1/2

for −∞ < x <∞, 0 0, the degree of freedom parameter.

Usage

dT(x, n=1, log=FALSE)pT(x, n=1, log.p=FALSE, lower.tail=TRUE)varT(p, n=1, log.p=FALSE, lower.tail=TRUE)esT(p, n=1)

Arguments



n the value of the degree of freedom parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

References


146 TL

Examples

x=runif(10,min=0,max=1)dT(x)pT(x)varT(x)esT(x)

TL Tukey-Lambda distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Tukey-Lambda distribution dueto Tukey (1962) given by

VaRp(X) =pλ − (1− p)λ

λ,

ESp(X) =pλ+1 + (1− p)λ+1 − 1

pλ(λ+ 1)

for 0 0, the shape parameter.

Usage

varTL(p, lambda=1, log.p=FALSE, lower.tail=TRUE)esTL(p, lambda=1)

Arguments


lambda the value of the shape parameter, must be positive, the default is 1




Value


Author(s)

Saralees Nadarajah

TL2 147

References


Examples

x=runif(10,min=0,max=1)varTL(x)esTL(x)

TL2 Topp-Leone distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Topp-Leone distribution due toTopp and Leone (1955) given by

f(x) = 2b(x(2− x))b−1(1− x),

F (x) = (x(2− x))b,

VaRp(X) = 1−√

1− p1/b,

ESp(X) = 1− b

pBp1/b

(b,

3

2

)for x > 0, 0 0, the shape parameter.

Usage

dTL2(x, b=1, log=FALSE)pTL2(x, b=1, log.p=FALSE, lower.tail=TRUE)varTL2(p, b=1, log.p=FALSE, lower.tail=TRUE)esTL2(p, b=1)

Arguments







Value


148 triangular

Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dTL2(x)pTL2(x)varTL2(x)esTL2(x)

triangular Triangular distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the triangular distribution given by

f(x) =

0, if x < a,

2(x− a)

(b− a)(c− a), if a ≤ x ≤ c,

2(b− x)

(b− a)(b− c), if c < x ≤ b,

0, if b < x,

F (x) =

0, if x < a,

(x− a)2

(b− a)(c− a), if a ≤ x ≤ c,

1− (b− x)2

(b− a)(b− c), if c < x ≤ b,

1, if b < x,

VaRp(X) =

a+

√p(b− a)(c− a), if 0 < p < c−a

b−a ,

b−√

(1− p)(b− a)(b− c), if c−ab−a ≤ p < 1,

ESp(X) =

a+

2

3

√p(b− a)(c− a), if 0 < p < c−a

b−a ,

b+a− cp

+2(2c− a− b)

3p+ 2√

(b− a)(b− c) (1− p)3/2

3p, if c−ab−a ≤ p < 1

triangular 149

for a ≤ x ≤ b, 0 < p < 1, −∞ < a < ∞, the first location parameter, −∞ < a < c < ∞, thesecond location parameter, and −∞ < c < b <∞, the third location parameter.

Usage

dtriangular(x, a=0, b=2, c=1, log=FALSE)ptriangular(x, a=0, b=2, c=1, log.p=FALSE, lower.tail=TRUE)vartriangular(p, a=0, b=2, c=1, log.p=FALSE, lower.tail=TRUE)estriangular(p, a=0, b=2, c=1)

Arguments




c the value of the second location parameter, can take any real value but must begreater than a, the default is 1

b the value of the third location parameter, can take any real value but must begreater than c, the default is 2




Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dtriangular(x)ptriangular(x)vartriangular(x)estriangular(x)

150 tsp

tsp Two sided power distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the two sided power distribution dueto van Dorp and Kotz (2002) given by

f(x) =

a(xθ

)a−1

, if 0 < x ≤ θ,

a

(1− x1− θ

)a−1

, if θ < x < 1,

F (x) =

θ(xθ

)a, if 0 < x ≤ θ,

1− (1− θ)(

1− x1− θ

)a, if θ < x < 1,

VaRp(X) =

θ(pθ

)1/a

, if 0 < p ≤ θ,

1− (1− θ)(

1− p1− θ

)1/a

, if θ 0, the shape parameter, and −∞ < θ <∞, the location parameter.

Usage

dtsp(x, a=1, theta=0.5, log=FALSE)ptsp(x, a=1, theta=0.5, log.p=FALSE, lower.tail=TRUE)vartsp(p, a=1, theta=0.5, log.p=FALSE, lower.tail=TRUE)estsp(p, a=1, theta=0.5)

Arguments



theta the value of the location parameter, must take a value in the unit interval, thedefault is 0.5





uniform 151

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dtsp(x)ptsp(x)vartsp(x)estsp(x)

uniform Uniform distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the uniform distribution given by

f(x) =1

b− a,

F (x) =x− ab− a

,

VaRp(X) = a+ p(b− a),

ESp(X) = a+p

2(b− a)

for a < x < b, 0 < p < 1, −∞ < a < ∞ , the first location parameter, and −∞ < a < b < ∞ ,the second location parameter.

Usage

duniform(x, a=0, b=1, log=FALSE)puniform(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)varuniform(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)esuniform(p, a=0, b=1)

152 weibull

Arguments


to be computeda the value of the first location parameter, can take any real value, the default is

zerob the value of the second location parameter, can take any real value but must be

greater than a, the default is 1log if TRUE then log(pdf) are returnedlog.p if TRUE then log(cdf) are returned and quantiles are computed for exp(p)lower.tail if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)duniform(x)puniform(x)varuniform(x)esuniform(x)

weibull Weibull distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Weibull distribution due toWeibull (1951) given by

f(x) =αxα−1

σαexp

{−(xσ

)α},

F (x) = 1− exp{−(xσ

)α},

VaRp(X) = σ [− log(1− p)]1/α ,ESp(X) =

σ

pγ (1 + 1/α,− log(1− p))

weibull 153

for x > 0, 0 0, the shape parameter, and σ > 0, the scale parameter.

Usage

dWeibull(x, alpha=1, sigma=1, log=FALSE)pWeibull(x, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)varWeibull(p, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)esWeibull(p, alpha=1, sigma=1)

Arguments








Value


Author(s)

Saralees Nadarajah

References


Examples

x=runif(10,min=0,max=1)dWeibull(x)pWeibull(x)varWeibull(x)esWeibull(x)

154 xie

xie Xie distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Xie distribution due to Xie et al.(2002) given by

f(x) = λb(xa

)b−1

exp[(x/a)b

]exp (λa) exp

{−λa exp

[(x/a)b

]},

F (x) = 1− exp (λa) exp{−λa exp

[(x/a)b

]},

VaRp(X) = a

{log

[1− log(1− p)

λa

]}1/b

,

ESp(X) =a

p

∫ p

0

{log

[1− log(1− v)

λa

]}1/b

dv

for x > 0, 0 0, the first scale parameter, b > 0, the shape parameter, and λ > 0, thesecond scale parameter.

Usage

dxie(x, a=1, b=1, lambda=1, log=FALSE)pxie(x, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)varxie(p, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)esxie(p, a=1, b=1, lambda=1)

Arguments









Value


Author(s)

Saralees Nadarajah

xie 155

References


Examples

x=runif(10,min=0,max=1)dxie(x)pxie(x)varxie(x)esxie(x)

Index

∗Topic Value at risk, expectedshortfall

aep, 4arcsine, 6ast, 8asylaplace, 9asypower, 11beard, 13betaburr, 15betaburr7, 16betadist, 17betaexp, 19betafrechet, 20betagompertz, 21betagumbel, 23betagumbel2, 24betalognorm, 25betalomax, 26betanorm, 28betapareto, 29betaweibull, 30BS, 32burr, 33burr7, 34Cauchy, 35chen, 37clg, 38compbeta, 39dagum, 41dweibull, 42expexp, 44expext, 45expgeo, 46explog, 47explogis, 49exponential, 50exppois, 51exppower, 52expweibull, 54

F, 55FR, 56frechet, 57Gamma, 59genbeta, 60genbeta2, 61geninvbeta, 63genlogis, 64genlogis3, 65genlogis4, 67genpareto, 68genpowerweibull, 69genunif, 71gev, 72gompertz, 73gumbel, 75gumbel2, 76halfcauchy, 77halflogis, 78halfnorm, 80halfT, 81HBlaplace, 82HL, 83Hlogis, 84invbeta, 86invexpexp, 87invgamma, 88kum, 89kumburr7, 91kumexp, 92kumgamma, 93kumgumbel, 95kumhalfnorm, 96kumloglogis, 97kumnormal, 99kumpareto, 100kumweibull, 102laplace, 103lfr, 104

156

INDEX 157

LNbeta, 106logbeta, 107logcauchy, 108loggamma, 109logisexp, 111logisrayleigh, 112logistic, 113loglaplace, 114loglog, 116loglogis, 117lognorm, 119lomax, 120Mlaplace, 121moexp, 123moweibull, 124MRbeta, 125nakagami, 127normal, 128pareto, 129paretostable, 130PCTAlaplace, 132perks, 133power1, 134power2, 136quad, 137rgamma, 138RS, 140schabe, 141secant, 142stacygamma, 143T, 144TL, 146TL2, 147triangular, 148tsp, 150uniform, 151weibull, 152xie, 154

∗Topic packageVaRES-package, 4

aep, 4arcsine, 6ast, 8asylaplace, 9asypower, 11

beard, 13betaburr, 15

betaburr7, 16betadist, 17betaexp, 19betafrechet, 20betagompertz, 21betagumbel, 23betagumbel2, 24betalognorm, 25betalomax, 26betanorm, 28betapareto, 29betaweibull, 30BS, 32burr, 33burr7, 34

Cauchy, 35chen, 37clg, 38compbeta, 39

daep (aep), 4dagum, 41darcsine (arcsine), 6dast (ast), 8dasylaplace (asylaplace), 9dasypower (asypower), 11dbeard (beard), 13dbetaburr (betaburr), 15dbetaburr7 (betaburr7), 16dbetadist (betadist), 17dbetaexp (betaexp), 19dbetafrechet (betafrechet), 20dbetagompertz (betagompertz), 21dbetagumbel (betagumbel), 23dbetagumbel2 (betagumbel2), 24dbetalognorm (betalognorm), 25dbetalomax (betalomax), 26dbetanorm (betanorm), 28dbetapareto (betapareto), 29dbetaweibull (betaweibull), 30dBS (BS), 32dburr (burr), 33dburr7 (burr7), 34dCauchy (Cauchy), 35dchen (chen), 37dclg (clg), 38dcompbeta (compbeta), 39ddagum (dagum), 41

158 INDEX

ddweibull (dweibull), 42dexpexp (expexp), 44dexpext (expext), 45dexpgeo (expgeo), 46dexplog (explog), 47dexplogis (explogis), 49dexponential (exponential), 50dexppois (exppois), 51dexppower (exppower), 52dexpweibull (expweibull), 54dF (F), 55dfrechet (frechet), 57dGamma (Gamma), 59dgenbeta (genbeta), 60dgenbeta2 (genbeta2), 61dgeninvbeta (geninvbeta), 63dgenlogis (genlogis), 64dgenlogis3 (genlogis3), 65dgenlogis4 (genlogis4), 67dgenpareto (genpareto), 68dgenpowerweibull (genpowerweibull), 69dgenunif (genunif), 71dgev (gev), 72dgompertz (gompertz), 73dgumbel (gumbel), 75dgumbel2 (gumbel2), 76dhalfcauchy (halfcauchy), 77dhalflogis (halflogis), 78dhalfnorm (halfnorm), 80dhalfT (halfT), 81dHBlaplace (HBlaplace), 82dHlogis (Hlogis), 84dinvbeta (invbeta), 86dinvexpexp (invexpexp), 87dinvgamma (invgamma), 88dkum (kum), 89dkumburr7 (kumburr7), 91dkumexp (kumexp), 92dkumgamma (kumgamma), 93dkumgumbel (kumgumbel), 95dkumhalfnorm (kumhalfnorm), 96dkumloglogis (kumloglogis), 97dkumnormal (kumnormal), 99dkumpareto (kumpareto), 100dkumweibull (kumweibull), 102dlaplace (laplace), 103dlfr (lfr), 104dLNbeta (LNbeta), 106

dlogbeta (logbeta), 107dlogcauchy (logcauchy), 108dloggamma (loggamma), 109dlogisexp (logisexp), 111dlogisrayleigh (logisrayleigh), 112dlogistic (logistic), 113dloglaplace (loglaplace), 114dloglog (loglog), 116dloglogis (loglogis), 117dlognorm (lognorm), 119dlomax (lomax), 120dMlaplace (Mlaplace), 121dmoexp (moexp), 123dmoweibull (moweibull), 124dMRbeta (MRbeta), 125dnakagami (nakagami), 127dnormal (normal), 128dpareto (pareto), 129dparetostable (paretostable), 130dPCTAlaplace (PCTAlaplace), 132dperks (perks), 133dpower1 (power1), 134dpower2 (power2), 136dquad (quad), 137drgamma (rgamma), 138dschabe (schabe), 141dsecant (secant), 142dstacygamma (stacygamma), 143dT (T), 144dTL2 (TL2), 147dtriangular (triangular), 148dtsp (tsp), 150duniform (uniform), 151dWeibull (weibull), 152dweibull, 42dxie (xie), 154

esaep (aep), 4esarcsine (arcsine), 6esast (ast), 8esasylaplace (asylaplace), 9esasypower (asypower), 11esbeard (beard), 13esbetaburr (betaburr), 15esbetaburr7 (betaburr7), 16esbetadist (betadist), 17esbetaexp (betaexp), 19esbetafrechet (betafrechet), 20esbetagompertz (betagompertz), 21

INDEX 159

esbetagumbel (betagumbel), 23esbetagumbel2 (betagumbel2), 24esbetalognorm (betalognorm), 25esbetalomax (betalomax), 26esbetanorm (betanorm), 28esbetapareto (betapareto), 29esbetaweibull (betaweibull), 30esBS (BS), 32esburr (burr), 33esburr7 (burr7), 34esCauchy (Cauchy), 35eschen (chen), 37esclg (clg), 38escompbeta (compbeta), 39esdagum (dagum), 41esdweibull (dweibull), 42esexpexp (expexp), 44esexpext (expext), 45esexpgeo (expgeo), 46esexplog (explog), 47esexplogis (explogis), 49esexponential (exponential), 50esexppois (exppois), 51esexppower (exppower), 52esexpweibull (expweibull), 54esF (F), 55esFR (FR), 56esfrechet (frechet), 57esGamma (Gamma), 59esgenbeta (genbeta), 60esgenbeta2 (genbeta2), 61esgeninvbeta (geninvbeta), 63esgenlogis (genlogis), 64esgenlogis3 (genlogis3), 65esgenlogis4 (genlogis4), 67esgenpareto (genpareto), 68esgenpowerweibull (genpowerweibull), 69esgenunif (genunif), 71esgev (gev), 72esgompertz (gompertz), 73esgumbel (gumbel), 75esgumbel2 (gumbel2), 76eshalfcauchy (halfcauchy), 77eshalflogis (halflogis), 78eshalfnorm (halfnorm), 80eshalfT (halfT), 81esHBlaplace (HBlaplace), 82esHL (HL), 83

esHlogis (Hlogis), 84esinvbeta (invbeta), 86esinvexpexp (invexpexp), 87esinvgamma (invgamma), 88eskum (kum), 89eskumburr7 (kumburr7), 91eskumexp (kumexp), 92eskumgamma (kumgamma), 93eskumgumbel (kumgumbel), 95eskumhalfnorm (kumhalfnorm), 96eskumloglogis (kumloglogis), 97eskumnormal (kumnormal), 99eskumpareto (kumpareto), 100eskumweibull (kumweibull), 102eslaplace (laplace), 103eslfr (lfr), 104esLNbeta (LNbeta), 106eslogbeta (logbeta), 107eslogcauchy (logcauchy), 108esloggamma (loggamma), 109eslogisexp (logisexp), 111eslogisrayleigh (logisrayleigh), 112eslogistic (logistic), 113esloglaplace (loglaplace), 114esloglog (loglog), 116esloglogis (loglogis), 117eslognorm (lognorm), 119eslomax (lomax), 120esMlaplace (Mlaplace), 121esmoexp (moexp), 123esmoweibull (moweibull), 124esMRbeta (MRbeta), 125esnakagami (nakagami), 127esnormal (normal), 128espareto (pareto), 129esparetostable (paretostable), 130esPCTAlaplace (PCTAlaplace), 132esperks (perks), 133espower1 (power1), 134espower2 (power2), 136esquad (quad), 137esrgamma (rgamma), 138esRS (RS), 140esschabe (schabe), 141essecant (secant), 142esstacygamma (stacygamma), 143esT (T), 144esTL (TL), 146

160 INDEX

esTL2 (TL2), 147estriangular (triangular), 148estsp (tsp), 150esuniform (uniform), 151esWeibull (weibull), 152esxie (xie), 154expexp, 44expext, 45expgeo, 46explog, 47explogis, 49exponential, 50exppois, 51exppower, 52expweibull, 54

F, 55FR, 56frechet, 57

Gamma, 59genbeta, 60genbeta2, 61geninvbeta, 63genlogis, 64genlogis3, 65genlogis4, 67genpareto, 68genpowerweibull, 69genunif, 71gev, 72gompertz, 73gumbel, 75gumbel2, 76

halfcauchy, 77halflogis, 78halfnorm, 80halfT, 81HBlaplace, 82HL, 83Hlogis, 84

invbeta, 86invexpexp, 87invgamma, 88

kum, 89kumburr7, 91

kumexp, 92kumgamma, 93kumgumbel, 95kumhalfnorm, 96kumloglogis, 97kumnormal, 99kumpareto, 100kumweibull, 102

laplace, 103lfr, 104LNbeta, 106logbeta, 107logcauchy, 108loggamma, 109logisexp, 111logisrayleigh, 112logistic, 113loglaplace, 114loglog, 116loglogis, 117lognorm, 119lomax, 120

Mlaplace, 121moexp, 123moweibull, 124MRbeta, 125

nakagami, 127normal, 128

paep (aep), 4parcsine (arcsine), 6pareto, 129paretostable, 130past (ast), 8pasylaplace (asylaplace), 9pasypower (asypower), 11pbeard (beard), 13pbetaburr (betaburr), 15pbetaburr7 (betaburr7), 16pbetadist (betadist), 17pbetaexp (betaexp), 19pbetafrechet (betafrechet), 20pbetagompertz (betagompertz), 21pbetagumbel (betagumbel), 23pbetagumbel2 (betagumbel2), 24pbetalognorm (betalognorm), 25

INDEX 161

pbetalomax (betalomax), 26pbetanorm (betanorm), 28pbetapareto (betapareto), 29pbetaweibull (betaweibull), 30pBS (BS), 32pburr (burr), 33pburr7 (burr7), 34pCauchy (Cauchy), 35pchen (chen), 37pclg (clg), 38pcompbeta (compbeta), 39PCTAlaplace, 132pdagum (dagum), 41pdweibull (dweibull), 42perks, 133pexpexp (expexp), 44pexpext (expext), 45pexpgeo (expgeo), 46pexplog (explog), 47pexplogis (explogis), 49pexponential (exponential), 50pexppois (exppois), 51pexppower (exppower), 52pexpweibull (expweibull), 54pF (F), 55pfrechet (frechet), 57pGamma (Gamma), 59pgenbeta (genbeta), 60pgenbeta2 (genbeta2), 61pgeninvbeta (geninvbeta), 63pgenlogis (genlogis), 64pgenlogis3 (genlogis3), 65pgenlogis4 (genlogis4), 67pgenpareto (genpareto), 68pgenpowerweibull (genpowerweibull), 69pgenunif (genunif), 71pgev (gev), 72pgompertz (gompertz), 73pgumbel (gumbel), 75pgumbel2 (gumbel2), 76phalfcauchy (halfcauchy), 77phalflogis (halflogis), 78phalfnorm (halfnorm), 80phalfT (halfT), 81pHBlaplace (HBlaplace), 82pHlogis (Hlogis), 84pinvbeta (invbeta), 86pinvexpexp (invexpexp), 87

pinvgamma (invgamma), 88pkum (kum), 89pkumburr7 (kumburr7), 91pkumexp (kumexp), 92pkumgamma (kumgamma), 93pkumgumbel (kumgumbel), 95pkumhalfnorm (kumhalfnorm), 96pkumloglogis (kumloglogis), 97pkumnormal (kumnormal), 99pkumpareto (kumpareto), 100pkumweibull (kumweibull), 102plaplace (laplace), 103plfr (lfr), 104pLNbeta (LNbeta), 106plogbeta (logbeta), 107plogcauchy (logcauchy), 108ploggamma (loggamma), 109plogisexp (logisexp), 111plogisrayleigh (logisrayleigh), 112plogistic (logistic), 113ploglaplace (loglaplace), 114ploglog (loglog), 116ploglogis (loglogis), 117plognorm (lognorm), 119plomax (lomax), 120pMlaplace (Mlaplace), 121pmoexp (moexp), 123pmoweibull (moweibull), 124pMRbeta (MRbeta), 125pnakagami (nakagami), 127pnormal (normal), 128power1, 134power2, 136ppareto (pareto), 129pparetostable (paretostable), 130pPCTAlaplace (PCTAlaplace), 132pperks (perks), 133ppower1 (power1), 134ppower2 (power2), 136pquad (quad), 137prgamma (rgamma), 138pschabe (schabe), 141psecant (secant), 142pstacygamma (stacygamma), 143pT (T), 144pTL2 (TL2), 147ptriangular (triangular), 148ptsp (tsp), 150

162 INDEX

puniform (uniform), 151pWeibull (weibull), 152pxie (xie), 154

quad, 137

rgamma, 138RS, 140

schabe, 141secant, 142stacygamma, 143

T, 144TL, 146TL2, 147triangular, 148tsp, 150

uniform, 151

varaep (aep), 4vararcsine (arcsine), 6varast (ast), 8varasylaplace (asylaplace), 9varasypower (asypower), 11varbeard (beard), 13varbetaburr (betaburr), 15varbetaburr7 (betaburr7), 16varbetadist (betadist), 17varbetaexp (betaexp), 19varbetafrechet (betafrechet), 20varbetagompertz (betagompertz), 21varbetagumbel (betagumbel), 23varbetagumbel2 (betagumbel2), 24varbetalognorm (betalognorm), 25varbetalomax (betalomax), 26varbetanorm (betanorm), 28varbetapareto (betapareto), 29varbetaweibull (betaweibull), 30varBS (BS), 32varburr (burr), 33varburr7 (burr7), 34varCauchy (Cauchy), 35varchen (chen), 37varclg (clg), 38varcompbeta (compbeta), 39vardagum (dagum), 41vardweibull (dweibull), 42VaRES (VaRES-package), 4

VaRES-package, 4varexpexp (expexp), 44varexpext (expext), 45varexpgeo (expgeo), 46varexplog (explog), 47varexplogis (explogis), 49varexponential (exponential), 50varexppois (exppois), 51varexppower (exppower), 52varexpweibull (expweibull), 54varF (F), 55varFR (FR), 56varfrechet (frechet), 57varGamma (Gamma), 59vargenbeta (genbeta), 60vargenbeta2 (genbeta2), 61vargeninvbeta (geninvbeta), 63vargenlogis (genlogis), 64vargenlogis3 (genlogis3), 65vargenlogis4 (genlogis4), 67vargenpareto (genpareto), 68vargenpowerweibull (genpowerweibull), 69vargenunif (genunif), 71vargev (gev), 72vargompertz (gompertz), 73vargumbel (gumbel), 75vargumbel2 (gumbel2), 76varhalfcauchy (halfcauchy), 77varhalflogis (halflogis), 78varhalfnorm (halfnorm), 80varhalfT (halfT), 81varHBlaplace (HBlaplace), 82varHL (HL), 83varHlogis (Hlogis), 84varinvbeta (invbeta), 86varinvexpexp (invexpexp), 87varinvgamma (invgamma), 88varkum (kum), 89varkumburr7 (kumburr7), 91varkumexp (kumexp), 92varkumgamma (kumgamma), 93varkumgumbel (kumgumbel), 95varkumhalfnorm (kumhalfnorm), 96varkumloglogis (kumloglogis), 97varkumnormal (kumnormal), 99varkumpareto (kumpareto), 100varkumweibull (kumweibull), 102varlaplace (laplace), 103

INDEX 163

varlfr (lfr), 104varLNbeta (LNbeta), 106varlogbeta (logbeta), 107varlogcauchy (logcauchy), 108varloggamma (loggamma), 109varlogisexp (logisexp), 111varlogisrayleigh (logisrayleigh), 112varlogistic (logistic), 113varloglaplace (loglaplace), 114varloglog (loglog), 116varloglogis (loglogis), 117varlognorm (lognorm), 119varlomax (lomax), 120varMlaplace (Mlaplace), 121varmoexp (moexp), 123varmoweibull (moweibull), 124varMRbeta (MRbeta), 125varnakagami (nakagami), 127varnormal (normal), 128varpareto (pareto), 129varparetostable (paretostable), 130varPCTAlaplace (PCTAlaplace), 132varperks (perks), 133varpower1 (power1), 134varpower2 (power2), 136varquad (quad), 137varrgamma (rgamma), 138varRS (RS), 140varschabe (schabe), 141varsecant (secant), 142varstacygamma (stacygamma), 143varT (T), 144varTL (TL), 146varTL2 (TL2), 147vartriangular (triangular), 148vartsp (tsp), 150varuniform (uniform), 151varWeibull (weibull), 152varxie (xie), 154

weibull, 152

xie, 154

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Package ‘VaRES’ · Computes Value at risk and expected shortfall, two most popular measures of ﬁnancial risk, for over one hundred parametric distributions, including all commonly